diff --git a/changelog b/changelog index d61af40..29750e1 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20080420 tpd src/input/schaum13.input show Schaums-Axiom equivalence 20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence 20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence 20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence diff --git a/src/input/schaum13.input.pamphlet b/src/input/schaum13.input.pamphlet index eb1a6eb..c4a0351 100644 --- a/src/input/schaum13.input.pamphlet +++ b/src/input/schaum13.input.pamphlet @@ -29,7 +29,7 @@ $$ )set message auto off )clear all ---S 1 of 19 +--S 1 aa:=integrate(1/sqrt(a*x^2+b*x+c),x) --R --R @@ -60,6 +60,183 @@ aa:=integrate(1/sqrt(a*x^2+b*x+c),x) --R \|- a --R Type: Union(List Expression Integer,...) --E + +--S 2 +bb1:=1/sqrt(a)*log(2*sqrt(a)*sqrt(a*x^2+b*x+c)*2*a*x+b) +--R +--R +--------------+ +--R +-+ | 2 +--R log(4a x\|a \|a x + b x + c + b) +--R (2) ---------------------------------- +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 3 +bb2:=-1/sqrt(-a)*asin((2*a*x+b)/sqrt(b^2-4*a*c)) +--R +--R 2a x + b +--R asin(--------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R (3) - -------------------- +--R +---+ +--R \|- a +--R Type: Expression Integer +--E + +--S 4 +bb3:=1/sqrt(a)*asinh((2*a*x+b)/sqrt(4*a*c-b^2)) +--R +--R 2a x + b +--R asinh(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R (4) ------------------- +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 5 +cc1:=bb1-aa.1 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 +--R log(4a x\|a \|a x + b x + c + b) +--R + +--R - +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 6 +cc2:=bb1-aa.2 +--R +--R (6) +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a log(4a x\|a \|a x + b x + c + b) +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +--R - 2\|a atan(------------------------------------) +--R a x +--R / +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 7 +cc3:=bb2-aa.1 +--R +--R (7) +--R - +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +-+ 2a x + b +--R - \|a asin(--------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 8 +cc4:=bb2-aa.2 +--R +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c 2a x + b +--R - 2atan(------------------------------------) - asin(--------------) +--R a x +-----------+ +--R | 2 +--R \|- 4a c + b +--R (8) -------------------------------------------------------------------- +--R +---+ +--R \|- a +--R Type: Expression Integer +--E + +--S 9 +cc5:=bb3-aa.1 +--R +--R (9) +--R - +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2a x + b +--R asinh(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 10 14:280 Axiom cannot simplify these expressions +cc6:=bb3-aa.2 +--R +--R (10) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +---+ 2a x + b +--R - 2\|a atan(------------------------------------) + \|- a asinh(------------) +--R a x +---------+ +--R | 2 +--R \|4a c - b +--R ----------------------------------------------------------------------------- +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.281~~~~~$\displaystyle @@ -70,7 +247,7 @@ $$ <<*>>= )clear all ---S 2 of 19 +--S 11 aa:=integrate(x/sqrt(a*x^2+b*x+c),x) --R --R @@ -121,6 +298,220 @@ aa:=integrate(x/sqrt(a*x^2+b*x+c),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 12 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 13 +bb1:=sqrt(a*x^2+b*x+c)/a-b/(2*a)*t1.1 +--R +--R (3) +--R - +--R b +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +--R 2\|a \|a x + b x + c +--R / +--R +-+ +--R 2a\|a +--R Type: Expression Integer +--E + +--S 14 +bb2:=sqrt(a*x^2+b*x+c)/a-b/(2*a)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--------------+ +--R \|- a \|a x + b x + c - \|- a \|c +---+ | 2 +--R - b atan(------------------------------------) + \|- a \|a x + b x + c +--R a x +--R ------------------------------------------------------------------------ +--R +---+ +--R a\|- a +--R Type: Expression Integer +--E + +--S 15 +cc1:=bb1-aa.1 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 2 +--R (- 2b\|c \|a x + b x + c + b x + 2b c) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 2b\|c \|a x + b x + c + b x + 2b c) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +-+ +-+ +--R - 4c\|a \|a x + b x + c + (2b x + 4c)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R 4a\|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|a +--R Type: Expression Integer +--E + +--S 16 +cc2:=bb1-aa.2 +--R +--R (6) +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R (- 2b\|- a \|c \|a x + b x + c + (b x + 2b c)\|- a ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R (4b\|a \|c \|a x + b x + c + (- 2b x - 4b c)\|a ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 +---+ +-+ +-+ +--R - 4c\|- a \|a \|a x + b x + c + (2b x + 4c)\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +---+ +-+ +--R 4a\|- a \|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|- a \|a +--R Type: Expression Integer +--E + +--S 17 +cc3:=bb2-aa.1 +--R +--R (7) +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R (- 2b\|- a \|c \|a x + b x + c + (b x + 2b c)\|- a ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R (- 4b\|a \|c \|a x + b x + c + (2b x + 4b c)\|a ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 +---+ +-+ +-+ +--R - 4c\|- a \|a \|a x + b x + c + (2b x + 4c)\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +---+ +-+ +--R 4a\|- a \|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|- a \|a +--R Type: Expression Integer +--E + +--S 18 14:281 Axiom cannot simplify these expressions +cc4:=bb2-aa.2 +--R +--R +--------------+ +--R | 2 +-+ +--R - 2c\|a x + b x + c + (b x + 2c)\|c +--R (8) -------------------------------------- +--R +--------------+ +--R +-+ | 2 +--R 2a\|c \|a x + b x + c - a b x - 2a c +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.282~~~~~$\displaystyle @@ -132,7 +523,7 @@ $$ <<*>>= )clear all ---S 3 of 19 +--S 19 aa:=integrate(x^2/(a*x^2+b*x+c),x) --R --R @@ -177,7 +568,247 @@ aa:=integrate(x^2/(a*x^2+b*x+c),x) --R 2a \|4a c - b --R ] --R Type: Union(List Expression Integer,...) ---E +--E + +--S 20 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 21 +bb1:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.1 +--R +--R (3) +--R 2 +--R (- 4a c + 3b ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +--R (4a x - 6b)\|a \|a x + b x + c +--R / +--R 2 +-+ +--R 8a \|a +--R Type: Expression Integer +--E + +--S 22 +bb2:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R 2 \|- a \|a x + b x + c - \|- a \|c +--R (- 4a c + 3b )atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 +--R (2a x - 3b)\|- a \|a x + b x + c +--R / +--R 2 +---+ +--R 4a \|- a +--R Type: Expression Integer +--E + +--S 23 +cc1:=aa.1-bb1 +--R +--R (5) +--R +-----------+ +--R 2 | 2 +--R (4a c - 3b )\|- 4a c + b +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 +-+ +--R (8a c - 4b )\|a +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--------------+ +--R | 2 +-+ | 2 +--R (- 4a x + 6b)\|- 4a c + b \|a \|a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +-+ +--R (- 4b log(a x + b x + c) + 8a x)\|- 4a c + b \|a +--R / +--R +-----------+ +--R 2 | 2 +-+ +--R 8a \|- 4a c + b \|a +--R Type: Expression Integer +--E + +--S 24 +cc2:=aa.2-bb1 +--R +--R (6) +--R +---------+ +--R 2 | 2 +--R (4a c - 3b )\|4a c - b +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +---------+ +--R | 2 +--R 2 +-+ (2a x + b)\|4a c - b +--R (- 16a c + 8b )\|a atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--------------+ +--R +-+ | 2 | 2 +--R (- 4a x + 6b)\|a \|4a c - b \|a x + b x + c +--R + +--R +---------+ +--R 2 +-+ | 2 +--R (- 4b log(a x + b x + c) + 8a x)\|a \|4a c - b +--R / +--R +---------+ +--R 2 +-+ | 2 +--R 8a \|a \|4a c - b +--R Type: Expression Integer +--E + +--S 25 +cc3:=aa.2-bb1 +--R +--R (7) +--R +---------+ +--R 2 | 2 +--R (4a c - 3b )\|4a c - b +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +---------+ +--R | 2 +--R 2 +-+ (2a x + b)\|4a c - b +--R (- 16a c + 8b )\|a atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--------------+ +--R +-+ | 2 | 2 +--R (- 4a x + 6b)\|a \|4a c - b \|a x + b x + c +--R + +--R +---------+ +--R 2 +-+ | 2 +--R (- 4b log(a x + b x + c) + 8a x)\|a \|4a c - b +--R / +--R +---------+ +--R 2 +-+ | 2 +--R 8a \|a \|4a c - b +--R Type: Expression Integer +--E + +--S 26 14:282 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (8) +--R +--------------+ +--R +---------+ +---+ | 2 +---+ +-+ +--R 2 | 2 \|- a \|a x + b x + c - \|- a \|c +--R (4a c - 3b )\|4a c - b atan(------------------------------------) +--R a x +--R + +--R +---------+ +--R | 2 +--R 2 +---+ (2a x + b)\|4a c - b +--R (- 8a c + 4b )\|- a atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--------------+ +--R +---+ | 2 | 2 +--R (- 2a x + 3b)\|- a \|4a c - b \|a x + b x + c +--R + +--R +---------+ +--R 2 +---+ | 2 +--R (- 2b log(a x + b x + c) + 4a x)\|- a \|4a c - b +--R / +--R +---------+ +--R 2 +---+ | 2 +--R 4a \|- a \|4a c - b +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.283~~~~~$\displaystyle @@ -200,7 +831,7 @@ $$ <<*>>= )clear all ---S 4 of 19 +--S 27 aa:=integrate(1/x*(a*x^2+b*x+c),x) --R --R @@ -210,6 +841,91 @@ aa:=integrate(1/x*(a*x^2+b*x+c),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 28 +bb1:=-1/sqrt(c)*log((2*sqrt(c)*sqrt(a*x^2+b*x+c)+b*x+2*c)/x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c + b x + 2c +--R log(---------------------------------) +--R x +--R (2) - -------------------------------------- +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 29 +bb2:=1/sqrt(-c)*asin((b*x+2*c)/(x*sqrt(b^2-4*a*c))) +--R +--R b x + 2c +--R asin(---------------) +--R +-----------+ +--R | 2 +--R x\|- 4a c + b +--R (3) --------------------- +--R +---+ +--R \|- c +--R Type: Expression Integer +--E + +--S 30 +bb3:=-1/sqrt(c)*asinh((b*x+2*c)/(x*sqrt(b^2-4*a*c))) +--R +--R b x + 2c +--R asinh(---------------) +--R +-----------+ +--R | 2 +--R x\|- 4a c + b +--R (4) - ---------------------- +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 31 +cc1:=aa-bb1 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c + b x + 2c 2 +-+ +--R 2log(---------------------------------) + (2c log(x) + a x + 2b x)\|c +--R x +--R ----------------------------------------------------------------------- +--R +-+ +--R 2\|c +--R Type: Expression Integer +--E + +--S 32 +cc2:=aa-bb2 +--R +--R b x + 2c 2 +---+ +--R - 2asin(---------------) + (2c log(x) + a x + 2b x)\|- c +--R +-----------+ +--R | 2 +--R x\|- 4a c + b +--R (6) ---------------------------------------------------------- +--R +---+ +--R 2\|- c +--R Type: Expression Integer +--E + +--S 33 14:283 Axiom cannot simplify these expressions +cc3:=aa-bb3 +--R +--R b x + 2c 2 +-+ +--R 2asinh(---------------) + (2c log(x) + a x + 2b x)\|c +--R +-----------+ +--R | 2 +--R x\|- 4a c + b +--R (7) ------------------------------------------------------- +--R +-+ +--R 2\|c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.284~~~~~$\displaystyle @@ -220,7 +936,7 @@ $$ <<*>>= )clear all ---S 5 of 19 +--S 34 aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x) --R --R @@ -266,6 +982,93 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 35 +t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (2) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 36 +bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*x)*t1 +--R +--R +--------------+ +--R +-+ | 2 +--------------+ +--R 2\|c \|a x + b x + c - b x - 2c +-+ | 2 +--R - b c log(---------------------------------) - 2\|c \|a x + b x + c +--R x +--R (3) --------------------------------------------------------------------- +--R +-+ +--R 2c x\|c +--R Type: Expression Integer +--E + +--S 37 +cc1:=aa.1-bb1 +--R +--R (4) +--R 2 +--R (2a c - b )x +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 2 | 2 +--R (b x log(a x + b x + c) - 2b x log(x) - 2bb1 c x - 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 | 2 +--R 2c x\|- 4a c + b +--R Type: Expression Integer +--E + +--S 38 14:284 Axiom cannot simplify these expressions +cc2:=aa.2-bb +--R +--R (5) +--R +--------------+ +--R +---------+ +-+ | 2 +--R 2 | 2 2\|c \|a x + b x + c - b x - 2c +--R b c \|4a c - b log(---------------------------------) +--R x +--R + +--R +---------+ +--R | 2 +--R 2 +-+ (2a x + b)\|4a c - b +--R (- 4a c + 2b )x\|c atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--------------+ +--R +-+ | 2 | 2 +--R 2c\|c \|4a c - b \|a x + b x + c +--R + +--R +---------+ +--R 2 +-+ | 2 +--R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|c \|4a c - b +--R / +--R +---------+ +--R 2 +-+ | 2 +--R 2c x\|c \|4a c - b +--R Type: Expression Integer +--E @ \section{\cite{1}:14.285~~~~~$\displaystyle @@ -277,7 +1080,7 @@ $$ <<*>>= )clear all ---S 6 of 19 +--S 39 aa:=integrate(sqrt(a*x^2+b*x+c),x) --R --R @@ -361,6 +1164,228 @@ aa:=integrate(sqrt(a*x^2+b*x+c),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 40 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 41 +bb1:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.1 +--R +--R (3) +--R 2 +--R (4a c - b ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +--R (4a x + 2b)\|a \|a x + b x + c +--R / +--R +-+ +--R 8a\|a +--R Type: Expression Integer +--E + +--S 42 +bb2:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R 2 \|- a \|a x + b x + c - \|- a \|c +--R (4a c - b )atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 +--R (2a x + b)\|- a \|a x + b x + c +--R / +--R +---+ +--R 4a\|- a +--R Type: Expression Integer +--E + +--S 43 +cc1:=aa.1-bb1 +--R +--R (5) +--R +--------------+ +--R 2 2 | 2 +--R (4b c x + 8b c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R ((- 4a b c - b )x - 8b c x - 8b c )\|c +--R / +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x +--R + +--R 2 +--R - 32a c +--R Type: Expression Integer +--E + +--S 44 +cc2:=aa.2-bb1 +--R +--R (6) +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((- 16a b c + 4b )x - 32a c + 8b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +---+ +--R ((16a c - b )x + (32a b c - 8b c)x + 32a c - 8b c )\|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((32a b c - 8b )x + 64a c - 16b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +-+ +--R ((- 32a c + 2b )x + (- 64a b c + 16b c)x - 64a c + 16b c )\|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (8b c x + 16b c )\|- a \|a \|a x + b x + c +--R + +--R 3 2 2 2 +---+ +-+ +-+ +--R ((- 8a b c - 2b )x - 16b c x - 16b c )\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a +--R Type: Expression Integer +--E + +--S 45 +cc3:=aa.1-bb2 +--R +--R (7) +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +---+ +--R ((- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c )\|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((- 32a b c + 8b )x - 64a c + 16b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +-+ +--R ((32a c - 2b )x + (64a b c - 16b c)x + 64a c - 16b c )\|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (8b c x + 16b c )\|- a \|a \|a x + b x + c +--R + +--R 3 2 2 2 +---+ +-+ +-+ +--R ((- 8a b c - 2b )x - 16b c x - 16b c )\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a +--R Type: Expression Integer +--E + +--S 46 14:285 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (8) +--R +--------------+ +--R 2 2 | 2 +--R (4b c x + 8b c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R ((- 4a b c - b )x - 8b c x - 8b c )\|c +--R / +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x +--R + +--R 2 +--R - 32a c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.286~~~~~$\displaystyle @@ -378,7 +1403,7 @@ $$ <<*>>= )clear all ---S 7 of 19 +--S 47 aa:=integrate(x*sqrt(a*x^2+b*x+c),x) --R --R @@ -522,6 +1547,416 @@ aa:=integrate(x*sqrt(a*x^2+b*x+c),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 48 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 49 +bb1:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.1 +--R +--R (3) +--R 3 +--R (- 12a b c + 3b ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 2 2 2 +-+ | 2 +--R (16a x + 4a b x + 16a c - 6b )\|a \|a x + b x + c +--R / +--R 2 +-+ +--R 48a \|a +--R Type: Expression Integer +--E + +--S 50 +bb2:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R 3 \|- a \|a x + b x + c - \|- a \|c +--R (- 12a b c + 3b )atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 2 +---+ | 2 +--R (8a x + 2a b x + 8a c - 3b )\|- a \|a x + b x + c +--R / +--R 2 +---+ +--R 24a \|- a +--R Type: Expression Integer +--E + +--S 51 +cc1:=aa.1-bb1 +--R +--R (5) +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 2 3 3 2 5 2 +--R (- 144a b c + 24a b c + 3b )x + (- 288a b c - 144a b c + 54b c)x +--R + +--R 2 3 4 2 4 3 3 +--R (- 576a b c + 144b c )x - 384a b c + 96b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 2 3 3 2 5 2 +--R (- 144a b c + 24a b c + 3b )x + (- 288a b c - 144a b c + 54b c)x +--R + +--R 2 3 4 2 4 3 3 +--R (- 576a b c + 144b c )x - 384a b c + 96b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 3 2 2 4 2 3 3 2 4 +--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c +--R + +--R 2 3 +--R - 192b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 768a b c + 288b c )x - 512a c + 192b c +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R 3 2 2 2 2 2 2 +-+ +-+ | 2 +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|a \|c \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 52 +cc2:=aa.2-bb1 +--R +--R (6) +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 144a b c + 24a b c + 3b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c )x +--R + +--R 4 3 3 +--R - 384a b c + 96b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 5 2 2 2 4 +--R (- 192a b c - 96a b c + 36b )x + (- 768a b c + 192b c)x +--R + +--R 3 3 2 +--R - 768a b c + 192b c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (288a b c - 48a b c - 6b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (576a b c + 288a b c - 108b c)x + (1152a b c - 288b c )x +--R + +--R 4 3 3 +--R 768a b c - 192b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 3 2 2 4 2 3 3 2 4 +--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c +--R + +--R 2 3 +--R - 192b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|a \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 768a b c + 288b c )x - 512a c + 192b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R / +--R 3 2 2 2 2 2 2 +---+ +-+ +-+ +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|- a \|a \|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 53 +cc3:=aa.1-bb2 +--R +--R (7) +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 144a b c + 24a b c + 3b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c )x +--R + +--R 4 3 3 +--R - 384a b c + 96b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 5 2 2 2 4 +--R (192a b c + 96a b c - 36b )x + (768a b c - 192b c)x +--R + +--R 3 3 2 +--R 768a b c - 192b c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 288a b c + 48a b c + 6b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 576a b c - 288a b c + 108b c)x + (- 1152a b c + 288b c )x +--R + +--R 4 3 3 +--R - 768a b c + 192b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 3 2 2 4 2 3 3 2 4 +--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c +--R + +--R 2 3 +--R - 192b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|a \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 768a b c + 288b c )x - 512a c + 192b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R / +--R 3 2 2 2 2 2 2 +---+ +-+ +-+ +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|- a \|a \|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 54 14:286 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (8) +--R 2 3 2 2 4 2 3 3 2 4 +--R (64a c + 24a b c - 18b c)x + (256a b c - 96b c )x + 256a c +--R + +--R 2 3 +--R - 96b c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 96a b c + 28a b c + 3b )x + (- 192a c - 72a b c + 54b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 384a b c + 144b c )x - 256a c + 96b c +--R * +--R +-+ +--R \|c +--R / +--R +--------------+ +--R 3 2 2 2 2 2 2 +-+ | 2 +--R ((192a c + 144a b )x + 768a b c x + 768a c )\|c \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 2 3 +--R (- 288a b c - 24a b )x + (- 576a c - 432a b c)x - 1152a b c x - 768a c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.287~~~~~$\displaystyle @@ -533,7 +1968,7 @@ $$ <<*>>= )clear all ---S 8 of 19 +--S 55 aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x) --R --R @@ -763,7 +2198,891 @@ aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x) --R \|- a --R ] --R Type: Union(List Expression Integer,...) ---E +--E + +--S 56 +t1:=integrate(sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 3 2 2 2 +-+ +--R ((- 16a c - 4a b )x + (- 40a b c - 2b )x + (- 32a c - 8b c)x)\|a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 +--R 16a b x + (32a c + 24a b )x + (56a b c + 6b )x +--R + +--R 2 2 +--R (32a c + 8b c)x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a +--R , +--R +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 2 3 3 2 2 2 +---+ +--R ((- 8a c - 2a b )x + (- 20a b c - b )x + (- 16a c - 4b c)x)\|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 2 2 +--R (8a b x + (16a c + 12a b )x + (28a b c + 3b )x + (16a c + 4b c)x) +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 +--R (16a b x + 32a c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +--R ((- 16a c - 4a b )x - 32a b c x - 32a c )\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 57 +bb1:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.1 +--R +--R (3) +--R 2 2 3 5 2 3 2 2 4 +--R ((- 192a b c + 288a b c - 60b )x - 384a c + 576a b c - 120b c) +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 3 3 2 2 2 4 6 2 +--R (192a c - 240a b c - 12a b c + 15b )x +--R + +--R 2 3 3 2 5 2 4 2 3 4 2 +--R (384a b c - 576a b c + 120b c)x + 384a c - 576a b c + 120b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 4 3 2 5 3 2 3 4 +--R (- 384a c - 96a b )x + (- 832a b c - 16a b )x +--R + +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (- 960a c - 96a b c + 20a b )x + (- 96a b c + 144a b c - 30b )x +--R + +--R 2 3 2 2 4 3 +--R (- 384a c + 896a b c - 120b c)x + 640a b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 4 6 4 3 2 5 3 2 3 4 +--R 384a b x + (768a c + 448a b )x + (1472a b c - 16a b )x +--R + +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (1152a c - 192a b c + 40a b )x + (- 32a b c - 512a b c + 90b )x +--R + +--R 2 3 2 2 4 3 +--R (384a c - 1216a b c + 120b c)x - 640a b c +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R 3 3 +-+ +-+ | 2 +--R (1536a b x + 3072a c)\|a \|c \|a x + b x + c +--R + +--R 4 3 2 2 3 3 2 +-+ +--R ((- 1536a c - 384a b )x - 3072a b c x - 3072a c )\|a +--R Type: Expression Integer +--E + +--S 58 +bb2:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.2 +--R +--R (4) +--R 2 2 3 5 2 3 2 2 4 +--R ((- 192a b c + 288a b c - 60b )x - 384a c + 576a b c - 120b c) +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 3 3 2 2 2 4 6 2 +--R (192a c - 240a b c - 12a b c + 15b )x +--R + +--R 2 3 3 2 5 2 4 2 3 4 2 +--R (384a b c - 576a b c + 120b c)x + 384a c - 576a b c + 120b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 4 3 2 5 3 2 3 4 +--R (- 192a c - 48a b )x + (- 416a b c - 8a b )x +--R + +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (- 480a c - 48a b c + 10a b )x + (- 48a b c + 72a b c - 15b )x +--R + +--R 2 3 2 2 4 3 +--R (- 192a c + 448a b c - 60b c)x + 320a b c +--R * +--R +--------------+ +--R +---+ | 2 +--R \|- a \|a x + b x + c +--R + +--R 4 6 4 3 2 5 3 2 3 4 +--R 192a b x + (384a c + 224a b )x + (736a b c - 8a b )x +--R + +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (576a c - 96a b c + 20a b )x + (- 16a b c - 256a b c + 45b )x +--R + +--R 2 3 2 2 4 3 +--R (192a c - 608a b c + 60b c)x - 320a b c +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R +--------------+ +--R 3 3 +---+ +-+ | 2 +--R (768a b x + 1536a c)\|- a \|c \|a x + b x + c +--R + +--R 4 3 2 2 3 3 2 +---+ +--R ((- 768a c - 192a b )x - 1536a b c x - 1536a c )\|- a +--R Type: Expression Integer +--E + +--S 59 +cc1:=aa.1-bb1 +--R +--R (5) +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (18432a c + 36864a b c - 77568a b c + 4200b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (98304a c + 73728a b c - 301056a b c + 69120b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (245760a b c - 368640a b c + 76800b c )x + 98304a c +--R + +--R 2 6 4 5 +--R - 147456a b c + 30720b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 8 +--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a b c +--R + +--R 10 +--R - 15b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 9 5 +--R (- 55296a b c + 36864a b c + 48384a b c - 9216a b c - 1080b c)x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c +--R + +--R 2 7 4 6 +--R 147456a b c - 30720b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (18432a c + 36864a b c - 77568a b c + 4200b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (98304a c + 73728a b c - 301056a b c + 69120b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (245760a b c - 368640a b c + 76800b c )x + 98304a c +--R + +--R 2 6 4 5 +--R - 147456a b c + 30720b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 8 +--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a b c +--R + +--R 10 +--R - 15b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 9 5 +--R (- 55296a b c + 36864a b c + 48384a b c - 9216a b c - 1080b c)x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c +--R + +--R 2 7 4 6 +--R 147456a b c - 30720b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (- 15360a b c - 12800a b c - 960a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (- 30720a b c - 107520a b c - 22400a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (- 245760a b c - 143360a b c )x + (- 163840a b c - 368640a b c )x +--R + +--R 2 6 7 +--R - 409600a b c x - 163840a b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 4 4 3 3 3 2 5 2 7 6 +--R (5120a b c + 19200a b c + 4800a b c + 80a b c)x +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (92160a b c + 76800a b c + 5760a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (92160a b c + 322560a b c + 67200a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (491520a b c + 286720a b c )x + (245760a b c + 552960a b c )x +--R + +--R 2 6 7 +--R 491520a b c x + 163840a b c +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R 5 2 4 3 3 5 5 +--R (73728a b c + 61440a b c + 4608a b )x +--R + +--R 5 3 4 2 2 3 4 4 +--R (147456a c + 516096a b c + 107520a b c)x +--R + +--R 4 3 3 3 2 3 4 4 3 2 3 2 +--R (1179648a b c + 688128a b c )x + (786432a c + 1769472a b c )x +--R + +--R 3 4 3 5 +--R 1966080a b c x + 786432a c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 6 3 5 2 2 4 4 3 6 6 +--R (- 24576a c - 92160a b c - 23040a b c - 384a b )x +--R + +--R 5 3 4 3 2 3 5 5 +--R (- 442368a b c - 368640a b c - 27648a b c)x +--R + +--R 5 4 4 2 3 3 4 2 4 +--R (- 442368a c - 1548288a b c - 322560a b c )x +--R + +--R 4 4 3 3 3 3 +--R (- 2359296a b c - 1376256a b c )x +--R + +--R 4 5 3 2 4 2 3 5 3 6 +--R (- 1179648a c - 2654208a b c )x - 2359296a b c x - 786432a c +--R * +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 60 +cc2:=aa.2-bb1 +--R +--R (6) +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (18432a c + 36864a b c - 77568a b c + 4200b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (98304a c + 73728a b c - 301056a b c + 69120b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (245760a b c - 368640a b c + 76800b c )x + 98304a c +--R + +--R 2 6 4 5 +--R - 147456a b c + 30720b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 8 +--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a b c +--R + +--R 10 +--R - 15b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 +--R - 55296a b c + 36864a b c + 48384a b c - 9216a b c +--R + +--R 9 +--R - 1080b c +--R * +--R 5 +--R x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c +--R + +--R 2 7 4 6 +--R 147456a b c - 30720b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (- 18432a b c + 12288a b c + 16128a b c - 3072a b c - 360b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (- 36864a c - 73728a b c + 155136a b c - 8400b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (- 196608a c - 147456a b c + 602112a b c - 138240b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (- 491520a b c + 737280a b c - 153600b c )x - 196608a c +--R + +--R 2 6 4 5 +--R 294912a b c - 61440b c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 8 +--R 6144a c + 13824a b c - 26880a b c - 1344a b c + 1656a b c +--R + +--R 10 +--R 30b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 +--R 110592a b c - 73728a b c - 96768a b c + 18432a b c +--R + +--R 9 +--R 2160b c +--R * +--R 5 +--R x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (110592a c + 221184a b c - 465408a b c + 25200b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (589824a b c - 540672a b c - 331776a b c + 107520b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (294912a c + 221184a b c - 903168a b c + 207360b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (589824a b c - 884736a b c + 184320b c )x + 196608a c +--R + +--R 2 7 4 6 +--R - 294912a b c + 61440b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (- 15360a b c - 12800a b c - 960a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (- 30720a b c - 107520a b c - 22400a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (- 245760a b c - 143360a b c )x + (- 163840a b c - 368640a b c )x +--R + +--R 2 6 7 +--R - 409600a b c x - 163840a b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|a \|a x + b x + c +--R + +--R 4 4 3 3 3 2 5 2 7 6 +--R (5120a b c + 19200a b c + 4800a b c + 80a b c)x +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (92160a b c + 76800a b c + 5760a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (92160a b c + 322560a b c + 67200a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (491520a b c + 286720a b c )x + (245760a b c + 552960a b c )x +--R + +--R 2 6 7 +--R 491520a b c x + 163840a b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R / +--R 5 2 4 3 3 5 5 +--R (73728a b c + 61440a b c + 4608a b )x +--R + +--R 5 3 4 2 2 3 4 4 +--R (147456a c + 516096a b c + 107520a b c)x +--R + +--R 4 3 3 3 2 3 4 4 3 2 3 2 +--R (1179648a b c + 688128a b c )x + (786432a c + 1769472a b c )x +--R + +--R 3 4 3 5 +--R 1966080a b c x + 786432a c +--R * +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R \|- a \|a \|c \|a x + b x + c +--R + +--R 6 3 5 2 2 4 4 3 6 6 +--R (- 24576a c - 92160a b c - 23040a b c - 384a b )x +--R + +--R 5 3 4 3 2 3 5 5 +--R (- 442368a b c - 368640a b c - 27648a b c)x +--R + +--R 5 4 4 2 3 3 4 2 4 +--R (- 442368a c - 1548288a b c - 322560a b c )x +--R + +--R 4 4 3 3 3 3 +--R (- 2359296a b c - 1376256a b c )x +--R + +--R 4 5 3 2 4 2 3 5 3 6 +--R (- 1179648a c - 2654208a b c )x - 2359296a b c x - 786432a c +--R * +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 61 +cc3:=aa.1-bb2 +--R +--R (7) +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (18432a c + 36864a b c - 77568a b c + 4200b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (98304a c + 73728a b c - 301056a b c + 69120b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (245760a b c - 368640a b c + 76800b c )x + 98304a c +--R + +--R 2 6 4 5 +--R - 147456a b c + 30720b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 8 +--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a b c +--R + +--R 10 +--R - 15b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 +--R - 55296a b c + 36864a b c + 48384a b c - 9216a b c +--R + +--R 9 +--R - 1080b c +--R * +--R 5 +--R x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c +--R + +--R 2 7 4 6 +--R 147456a b c - 30720b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (18432a b c - 12288a b c - 16128a b c + 3072a b c + 360b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (36864a c + 73728a b c - 155136a b c + 8400b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (294912a b c - 270336a b c - 165888a b c + 53760b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (196608a c + 147456a b c - 602112a b c + 138240b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (491520a b c - 737280a b c + 153600b c )x + 196608a c +--R + +--R 2 6 4 5 +--R - 294912a b c + 61440b c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 +--R - 6144a c - 13824a b c + 26880a b c + 1344a b c +--R + +--R 8 10 +--R - 1656a b c - 30b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 +--R - 110592a b c + 73728a b c + 96768a b c - 18432a b c +--R + +--R 9 +--R - 2160b c +--R * +--R 5 +--R x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (- 110592a c - 221184a b c + 465408a b c - 25200b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (- 589824a b c + 540672a b c + 331776a b c - 107520b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (- 294912a c - 221184a b c + 903168a b c - 207360b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (- 589824a b c + 884736a b c - 184320b c )x - 196608a c +--R + +--R 2 7 4 6 +--R 294912a b c - 61440b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (- 15360a b c - 12800a b c - 960a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (- 30720a b c - 107520a b c - 22400a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (- 245760a b c - 143360a b c )x + (- 163840a b c - 368640a b c )x +--R + +--R 2 6 7 +--R - 409600a b c x - 163840a b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|a \|a x + b x + c +--R + +--R 4 4 3 3 3 2 5 2 7 6 +--R (5120a b c + 19200a b c + 4800a b c + 80a b c)x +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (92160a b c + 76800a b c + 5760a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (92160a b c + 322560a b c + 67200a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (491520a b c + 286720a b c )x + (245760a b c + 552960a b c )x +--R + +--R 2 6 7 +--R 491520a b c x + 163840a b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R / +--R 5 2 4 3 3 5 5 +--R (73728a b c + 61440a b c + 4608a b )x +--R + +--R 5 3 4 2 2 3 4 4 +--R (147456a c + 516096a b c + 107520a b c)x +--R + +--R 4 3 3 3 2 3 4 4 3 2 3 2 +--R (1179648a b c + 688128a b c )x + (786432a c + 1769472a b c )x +--R + +--R 3 4 3 5 +--R 1966080a b c x + 786432a c +--R * +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R \|- a \|a \|c \|a x + b x + c +--R + +--R 6 3 5 2 2 4 4 3 6 6 +--R (- 24576a c - 92160a b c - 23040a b c - 384a b )x +--R + +--R 5 3 4 3 2 3 5 5 +--R (- 442368a b c - 368640a b c - 27648a b c)x +--R + +--R 5 4 4 2 3 3 4 2 4 +--R (- 442368a c - 1548288a b c - 322560a b c )x +--R + +--R 4 4 3 3 3 3 +--R (- 2359296a b c - 1376256a b c )x +--R + +--R 4 5 3 2 4 2 3 5 3 6 +--R (- 1179648a c - 2654208a b c )x - 2359296a b c x - 786432a c +--R * +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 62 14:287 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (8) +--R 2 2 4 4 3 6 2 5 +--R (- 960a b c - 800a b c - 60b c )x +--R + +--R 2 5 3 4 5 3 4 +--R (- 1920a b c - 6720a b c - 1400b c )x +--R + +--R 2 5 4 4 3 6 3 5 2 +--R (- 15360a b c - 8960b c )x + (- 10240a b c - 23040b c )x +--R + +--R 2 6 7 +--R - 25600b c x - 10240b c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 4 2 3 3 5 2 7 6 +--R (320a b c + 1200a b c + 300a b c + 5b c)x +--R + +--R 2 2 4 4 3 6 2 5 +--R (5760a b c + 4800a b c + 360b c )x +--R + +--R 2 5 3 4 5 3 4 2 5 4 4 3 +--R (5760a b c + 20160a b c + 4200b c )x + (30720a b c + 17920b c )x +--R + +--R 6 3 5 2 2 6 7 +--R (15360a b c + 34560b c )x + 30720b c x + 10240b c +--R * +--R +-+ +--R \|c +--R / +--R 4 2 3 3 2 5 5 +--R (4608a b c + 3840a b c + 288a b )x +--R + +--R 4 3 3 2 2 2 4 4 +--R (9216a c + 32256a b c + 6720a b c)x +--R + +--R 3 3 2 3 2 3 3 4 2 2 3 2 +--R (73728a b c + 43008a b c )x + (49152a c + 110592a b c )x +--R + +--R 2 4 2 5 +--R 122880a b c x + 49152a c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 5 3 4 2 2 3 4 2 6 6 +--R (- 1536a c - 5760a b c - 1440a b c - 24a b )x +--R + +--R 4 3 3 3 2 2 5 5 +--R (- 27648a b c - 23040a b c - 1728a b c)x +--R + +--R 4 4 3 2 3 2 4 2 4 +--R (- 27648a c - 96768a b c - 20160a b c )x +--R + +--R 3 4 2 3 3 3 3 5 2 2 4 2 +--R (- 147456a b c - 86016a b c )x + (- 73728a c - 165888a b c )x +--R + +--R 2 5 2 6 +--R - 147456a b c x - 49152a c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.288~~~~~$\displaystyle @@ -775,7 +3094,7 @@ $$ <<*>>= )clear all ---S 9 of 19 +--S 63 aa:=integrate(sqrt(a*x^2+b*x+c)/x,x) --R --R @@ -858,6 +3177,307 @@ aa:=integrate(sqrt(a*x^2+b*x+c)/x,x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 64 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 65 +t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (3) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 66 +bb1:=sqrt(a*x^2+b*x+c)+b/2*t1.1+c*t2 +--R +--R (4) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R 2c\|a log(---------------------------------) +--R x +--R + +--R +-+ +--R b\|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 +--R 2\|a \|c \|a x + b x + c +--R / +--R +-+ +-+ +--R 2\|a \|c +--R Type: Expression Integer +--E + +--S 67 +bb2:=sqrt(a*x^2+b*x+c)+b/2*t1.2+c*t2 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +--R c\|- a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +--R b\|c atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R / +--R +---+ +-+ +--R \|- a \|c +--R Type: Expression Integer +--E + +--S 68 +cc1:=aa.1-bb1 +--R +--R (6) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R - 2c\|a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R 2c\|a log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R - +--R +-+ +--R b\|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +-+ +--R b\|c +--R * +--R log +--R 2 +-+ 2 2 +-+ +--R ((- 2a b x - 8a c x)\|c + (4a c x + 4b c x + 8c )\|a ) +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 2 2 +-+ +-+ 2 3 +--R (- 2a b x + (- 8a c - b )x - 8b c x - 8c )\|a \|c + 4a c x +--R + +--R 2 2 +--R 6a b c x + 8a c x +--R / +--R +--------------+ +--R 2 | 2 +--R (4b c x + 8c )\|a x + b x + c +--R + +--R 2 2 2 +-+ +--R ((- 4a c - b )x - 8b c x - 8c )\|c +--R + +--R +-+ +--R 2c\|a +--R / +--R +-+ +-+ +--R 2\|a \|c +--R Type: Expression Integer +--E + +--S 69 +cc2:=aa.2-bb1 +--R +--R (7) +--R +--------------+ +--R +-+ | 2 +--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c +--R - 2c\|- a \|a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c +--R 2c\|- a \|a log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R - +--R +---+ +-+ +--R b\|- a \|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +---+ +-+ | 2 +---+ +--R +-+ +-+ \|- a \|c \|a x + b x + c - c\|- a +---+ +-+ +--R 2b\|a \|c atan(-------------------------------------) + 2c\|- a \|a +--R +-+ +--R a x\|c +--R / +--R +---+ +-+ +-+ +--R 2\|- a \|a \|c +--R Type: Expression Integer +--E + +--S 70 +cc3:=aa.1-bb2 +--R +--R (8) +--R +--------------+ +--R +-+ | 2 +--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c +--R - 2c\|- a \|a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c +--R 2c\|- a \|a log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R +---+ +-+ +--R b\|- a \|c +--R * +--R log +--R 2 +-+ 2 2 +-+ +--R ((- 2a b x - 8a c x)\|c + (4a c x + 4b c x + 8c )\|a ) +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 2 2 +-+ +-+ 2 3 +--R (- 2a b x + (- 8a c - b )x - 8b c x - 8c )\|a \|c + 4a c x +--R + +--R 2 2 +--R 6a b c x + 8a c x +--R / +--R +--------------+ +--R 2 | 2 +--R (4b c x + 8c )\|a x + b x + c +--R + +--R 2 2 2 +-+ +--R ((- 4a c - b )x - 8b c x - 8c )\|c +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ +-+ \|- a \|a x + b x + c - \|- a \|c +---+ +-+ +--R - 2b\|a \|c atan(------------------------------------) + 2c\|- a \|a +--R a x +--R / +--R +---+ +-+ +-+ +--R 2\|- a \|a \|c +--R Type: Expression Integer +--E + +--S 71 14:288 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (9) +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +--R - c\|- a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +--R c\|- a log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R +--------------+ +--R +---+ +-+ | 2 +---+ +--R +-+ \|- a \|c \|a x + b x + c - c\|- a +--R b\|c atan(-------------------------------------) +--R +-+ +--R a x\|c +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +---+ +--R - b\|c atan(------------------------------------) + c\|- a +--R a x +--R / +--R +---+ +-+ +--R \|- a \|c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.289~~~~~$\displaystyle @@ -870,7 +3490,7 @@ $$ <<*>>= )clear all ---S 10 of 19 +--S 72 aa:=integrate(sqrt(a*x^2+b*x+c)/x^2,x) --R --R @@ -937,7 +3557,345 @@ aa:=integrate(sqrt(a*x^2+b*x+c)/x^2,x) --R 8c x\|a x + b x + c + (- 4b x - 8c x)\|c --R ] --R Type: Union(List Expression Integer,...) ---E +--E + +--S 73 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 74 +t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (3) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 75 +bb1:=-sqrt(a*x^2+b*x+c)/x+a*t1.1+b/2*t2 +--R +--R (4) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R b x\|a log(---------------------------------) +--R x +--R + +--R +-+ +--R 2a x\|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 +--R - 2\|a \|c \|a x + b x + c +--R / +--R +-+ +-+ +--R 2x\|a \|c +--R Type: Expression Integer +--E + +--S 76 +bb2:=-sqrt(a*x^2+b*x+c)/x+a*t1.2+b/2*t2 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +--R b x\|- a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +--R 4a x\|c atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 +--R - 2\|- a \|c \|a x + b x + c +--R / +--R +---+ +-+ +--R 2x\|- a \|c +--R Type: Expression Integer +--E + +--S 77 +cc1:=aa.1-bb1 +--R +--R (6) +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R (- 4b c\|a \|a x + b x + c + (2b x + 4b c)\|a \|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R (4b c\|a \|a x + b x + c + (- 2b x - 4b c)\|a \|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 8a c\|c \|a x + b x + c + 4a b c x + 8a c ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (8a c\|c \|a x + b x + c - 4a b c x - 8a c ) +--R * +--R +--------------+ +--R +-+ +-+ | 2 +-+ +-+ 2 +--R (2\|c - 2x\|a )\|a x + b x + c + 2x\|a \|c - 2a x - b x - 2c +--R log(-----------------------------------------------------------------) +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R - 2b c\|a \|a x + b x + c + (b x + 2b c)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R 8c\|a \|c \|a x + b x + c + (- 4b c x - 8c )\|a +--R Type: Expression Integer +--E + +--S 78 +cc2:=aa.2-bb1 +--R +--R (7) +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R (- 4b c\|a \|a x + b x + c + (2b x + 4b c)\|a \|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R (4b c\|a \|a x + b x + c + (- 2b x - 4b c)\|a \|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 8a c\|c \|a x + b x + c + 4a b c x + 8a c ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +---+ +-+ +-+ | 2 2 +---+ +-+ +--R (16c\|- a \|a \|c \|a x + b x + c + (- 8b c x - 16c )\|- a \|a ) +--R * +--R +--------------+ +--R | 2 +-+ +--R \|a x + b x + c - \|c +--R atan(------------------------) +--R +---+ +--R x\|- a +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R - 2b c\|a \|a x + b x + c + (b x + 2b c)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R 8c\|a \|c \|a x + b x + c + (- 4b c x - 8c )\|a +--R Type: Expression Integer +--E + +--S 79 +cc3:=aa.1-bb2 +--R +--R (8) +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R (- 4b c\|- a \|a x + b x + c + (2b x + 4b c)\|- a \|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R (4b c\|- a \|a x + b x + c + (- 2b x - 4b c)\|- a \|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +---+ +-+ +-+ | 2 2 +---+ +-+ +--R (8c\|- a \|a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a \|a ) +--R * +--R +--------------+ +--R +-+ +-+ | 2 +-+ +-+ 2 +--R (2\|c - 2x\|a )\|a x + b x + c + 2x\|a \|c - 2a x - b x - 2c +--R log(-----------------------------------------------------------------) +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 16a c\|c \|a x + b x + c + 8a b c x + 16a c ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R - 2b c\|- a \|a x + b x + c + (b x + 2b c)\|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R 8c\|- a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a +--R Type: Expression Integer +--E + +--S 80 14:289 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (9) +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R (- 4b c\|- a \|a x + b x + c + (2b x + 4b c)\|- a \|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R (4b c\|- a \|a x + b x + c + (- 2b x - 4b c)\|- a \|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 16a c\|c \|a x + b x + c + 8a b c x + 16a c ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 16a c\|c \|a x + b x + c + 8a b c x + 16a c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R \|a x + b x + c - \|c +--R atan(------------------------) +--R +---+ +--R x\|- a +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R - 2b c\|- a \|a x + b x + c + (b x + 2b c)\|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R 8c\|- a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.290~~~~~$\displaystyle @@ -949,7 +3907,7 @@ $$ <<*>>= )clear all ---S 11 of 19 +--S 81 aa:=integrate(1/(a*x^2+b*x+c)^(3/2),x) --R --R @@ -962,6 +3920,31 @@ aa:=integrate(1/(a*x^2+b*x+c)^(3/2),x) --R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c --R Type: Union(Expression Integer,...) --E + +--S 82 +bb:=(2*(2*a*x+b))/((4*a*c-b^2)*sqrt(a*x^2+b*x+c)) +--R +--R 4a x + 2b +--R (2) ---------------------------- +--R +--------------+ +--R 2 | 2 +--R (4a c - b )\|a x + b x + c +--R Type: Expression Integer +--E + +--S 83 14:290 Axiom cannot simplify these expressions +cc:=aa-bb +--R +--R (3) +--R +--------------+ +--R +-+ | 2 2 +--R 4b\|c \|a x + b x + c - 2b x - 4b c +--R ----------------------------------------------------------------------- +--R +--------------+ +--R 2 2 | 2 3 2 2 +-+ +--R (8a c - 2b c)\|a x + b x + c + ((- 4a b c + b )x - 8a c + 2b c)\|c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.291~~~~~$\displaystyle @@ -972,7 +3955,7 @@ $$ <<*>>= )clear all ---S 12 of 19 +--S 84 aa:=integrate(x/(a*x^2+b*x+c)^(3/2),x) --R --R @@ -984,6 +3967,31 @@ aa:=integrate(x/(a*x^2+b*x+c)^(3/2),x) --R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c --R Type: Union(Expression Integer,...) --E + +--S 85 +bb:=(2*(b*x+2*c))/((b^2-4*a*c)*sqrt(a*x^2+b*x+c)) +--R +--R - 2b x - 4c +--R (2) ---------------------------- +--R +--------------+ +--R 2 | 2 +--R (4a c - b )\|a x + b x + c +--R Type: Expression Integer +--E + +--S 86 14:291 Axiom cannot simplify these expressions +cc:=aa-bb +--R +--R (3) +--R +--------------+ +--R +-+ | 2 2 +--R - 8c\|c \|a x + b x + c + 4b c x + 8c +--R ----------------------------------------------------------------------- +--R +--------------+ +--R 2 2 | 2 3 2 2 +-+ +--R (8a c - 2b c)\|a x + b x + c + ((- 4a b c + b )x - 8a c + 2b c)\|c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.292~~~~~$\displaystyle @@ -995,7 +4003,7 @@ $$ <<*>>= )clear all ---S 13 of 19 +--S 87 aa:=integrate(x^2/(a*x^2+b*x+c)^(3/2),x) --R --R @@ -1052,6 +4060,356 @@ aa:=integrate(x^2/(a*x^2+b*x+c)^(3/2),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 88 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 89 +bb1:=((2*b^2-4*a*c)*x+2*b*c)/(4*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.1 +--R +--R (3) +--R +--------------+ +--R 2 | 2 +--R (8a c - 2b )\|a x + b x + c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 +-+ +--R ((- 2a c + a b )x + a b c)\|a +--R / +--R +--------------+ +--R 2 2 +-+ | 2 +--R (8a c - 2a b )\|a \|a x + b x + c +--R Type: Expression Integer +--E + +--S 90 +bb2:=((2*b^2-4*a*c)*x+2*b*c)/(4*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.2 +--R +--R (4) +--R +--------------+ +--R +--------------+ +---+ | 2 +---+ +-+ +--R 2 | 2 \|- a \|a x + b x + c - \|- a \|c +--R (16a c - 4b )\|a x + b x + c atan(------------------------------------) +--R a x +--R + +--R 2 2 +---+ +--R ((- 2a c + a b )x + a b c)\|- a +--R / +--R +--------------+ +--R 2 2 +---+ | 2 +--R (8a c - 2a b )\|- a \|a x + b x + c +--R Type: Expression Integer +--E + +--S 91 +cc1:=aa.1-bb1 +--R +--R (5) +--R 2 3 2 2 2 2 +--R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b c)x +--R + +--R 2 +--R 2a b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 3 +--R ((4a - 16a )c + (- 2a + 4a)b c)x +--R + +--R 2 2 3 2 2 3 2 2 +--R ((2a - 16a)b c + (- 2a + 4)b c)x + ((4a - 16a)c + (- 4a + 4)b c )x +--R + +--R 3 +--R - 2a b c +--R / +--R 3 2 2 2 2 2 2 3 2 3 2 2 +--R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c ) +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 4 2 +--R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x +--R + +--R 2 2 3 2 3 2 2 +--R (- 24a b c + 6a b c)x - 16a c + 4a b c +--R * +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 92 +cc2:=aa.2-bb1 +--R +--R (6) +--R 2 2 2 2 2 3 3 2 2 +--R ((- 16a c + 4a b c)x + (- 16a b c + 4b c)x - 16a c + 4b c ) +--R * +--R +--------------+ +--R +---+ | 2 +--R \|- a \|a x + b x + c +--R + +--R 2 3 3 2 2 2 4 2 +--R (8a b c - 2a b )x + (16a c + 4a b c - 2b )x +--R + +--R 2 3 3 2 2 +--R (24a b c - 6b c)x + 16a c - 4b c +--R * +--R +---+ +-+ +--R \|- a \|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 2 2 2 3 3 2 2 +-+ +--R ((32a c - 8a b c)x + (32a b c - 8b c)x + 32a c - 8b c )\|a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 3 3 2 2 2 4 2 +--R (- 16a b c + 4a b )x + (- 32a c - 8a b c + 4b )x +--R + +--R 2 3 3 2 2 +--R (- 48a b c + 12b c)x - 32a c + 8b c +--R * +--R +-+ +-+ +--R \|a \|c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 3 2 2 2 2 +--R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b c)x +--R + +--R 2 +--R 2a b c +--R * +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R \|- a \|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 3 +--R ((4a - 16a )c + (- 2a + 4a)b c)x +--R + +--R 2 2 3 2 +--R ((2a - 16a)b c + (- 2a + 4)b c)x +--R + +--R 2 3 2 2 3 +--R ((4a - 16a)c + (- 4a + 4)b c )x - 2a b c +--R * +--R +---+ +-+ +--R \|- a \|a +--R / +--R 3 2 2 2 2 2 2 3 2 3 2 2 +---+ +--R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c )\|- a +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 4 2 +--R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x +--R + +--R 2 2 3 2 3 2 2 +--R (- 24a b c + 6a b c)x - 16a c + 4a b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R Type: Expression Integer +--E + +--S 93 +cc3:=aa.1-bb2 +--R +--R (7) +--R 2 2 2 2 2 3 3 2 2 +---+ +--R ((16a c - 4a b c)x + (16a b c - 4b c)x + 16a c - 4b c )\|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 3 3 2 2 2 4 2 +--R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2b )x +--R + +--R 2 3 3 2 2 +--R (- 24a b c + 6b c)x - 16a c + 4b c +--R * +--R +---+ +-+ +--R \|- a \|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 2 2 2 3 3 2 2 +-+ +--R ((- 32a c + 8a b c)x + (- 32a b c + 8b c)x - 32a c + 8b c )\|a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 3 3 2 2 2 4 2 +--R (16a b c - 4a b )x + (32a c + 8a b c - 4b )x +--R + +--R 2 3 3 2 2 +--R (48a b c - 12b c)x + 32a c - 8b c +--R * +--R +-+ +-+ +--R \|a \|c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 3 2 2 2 2 +--R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b c)x +--R + +--R 2 +--R 2a b c +--R * +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R \|- a \|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 3 +--R ((4a - 16a )c + (- 2a + 4a)b c)x +--R + +--R 2 2 3 2 +--R ((2a - 16a)b c + (- 2a + 4)b c)x +--R + +--R 2 3 2 2 3 +--R ((4a - 16a)c + (- 4a + 4)b c )x - 2a b c +--R * +--R +---+ +-+ +--R \|- a \|a +--R / +--R 3 2 2 2 2 2 2 3 2 3 2 2 +---+ +--R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c )\|- a +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 4 2 +--R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x +--R + +--R 2 2 3 2 3 2 2 +--R (- 24a b c + 6a b c)x - 16a c + 4a b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R Type: Expression Integer +--E + +--S 94 14:292 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (8) +--R 2 3 2 2 2 2 +--R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b c)x +--R + +--R 2 +--R 2a b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 3 +--R ((4a - 16a )c + (- 2a + 4a)b c)x +--R + +--R 2 2 3 2 2 3 2 2 +--R ((2a - 16a)b c + (- 2a + 4)b c)x + ((4a - 16a)c + (- 4a + 4)b c )x +--R + +--R 3 +--R - 2a b c +--R / +--R 3 2 2 2 2 2 2 3 2 3 2 2 +--R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c ) +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 4 2 +--R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x +--R + +--R 2 2 3 2 3 2 2 +--R (- 24a b c + 6a b c)x - 16a c + 4a b c +--R * +--R +-+ +--R \|c +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.293~~~~~$\displaystyle @@ -1064,7 +4422,7 @@ $$ <<*>>= )clear all ---S 14 of 19 +--S 95 aa:=integrate(1/(x*(a*x^2+b*x+c)^(3/2)),x) --R --R @@ -1089,6 +4447,110 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)^(3/2)),x) --R (b c x + 2c )\|c \|a x + b x + c - 2a c x - 2b c x - 2c --R Type: Union(Expression Integer,...) --E + +--S 96 +t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (2) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 97 +t2:=integrate(1/(a*x^2+b*x+c)^(3/2),x) +--R +--R +--------------+ +--R | 2 +-+ +--R - 2x\|a x + b x + c + 2x\|c +--R (3) -------------------------------------------------------- +--R +--------------+ +--R +-+ | 2 2 2 +--R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c +--R Type: Union(Expression Integer,...) +--E + +--S 98 +bb:=1/(c*sqrt(a*x^2+b*x+c))+1/c*t1-b/(2*c)*t2 +--R +--R (4) +--R +--------------+ +--R 2 2 | 2 +--R (2a c x + 2b c x + 2c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R (- a b x + (- 2a c - b )x - 3b c x - 2c )\|c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R 2 | 2 3 2 2 2 +-+ +--R - 2c \|a x + b x + c + (- a b x + (2a c - b )x + b c x + 2c )\|c +--R / +--R +--------------+ +--R 2 2 2 3 +-+ | 2 2 3 +--R (2a c x + 2b c x + 2c )\|c \|a x + b x + c - a b c x +--R + +--R 3 2 2 2 3 4 +--R (- 2a c - b c )x - 3b c x - 2c +--R Type: Expression Integer +--E + +--S 99 14:293 Axiom cannot simplify these expressions +cc:=aa-bb +--R +--R (5) +--R +--------------+ +--R 2 2 2 | 2 +--R ((- 4a c - b )x - 8b c x - 8c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R (4a b x + (8a c + 4b )x + 12b c x + 8c )\|c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R 2 2 2 | 2 +--R ((4a c + b )x + 8b c x + 8c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R (- 4a b x + (- 8a c - 4b )x - 12b c x - 8c )\|c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R +--------------+ +--R 2 2 2 | 2 +--R ((4a c + b )x + 8b c x + 8c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R (- 4a b x + (- 8a c - 4b )x - 12b c x - 8c )\|c +--R / +--R +--------------+ +--R 2 2 2 2 3 +-+ | 2 2 3 +--R ((4a c + b c)x + 8b c x + 8c )\|c \|a x + b x + c - 4a b c x +--R + +--R 3 2 2 2 3 4 +--R (- 8a c - 4b c )x - 12b c x - 8c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.294~~~~~$\displaystyle @@ -1105,7 +4567,7 @@ $$ <<*>>= )clear all ---S 15 of 19 +--S 100 aa:=integrate(1/(x^2*(a*x^2+b*x+c)^(3/2)),x) --R --R @@ -1145,6 +4607,149 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)^(3/2)),x) --R (- 16a b c x + (- 32a c - 16b c )x - 48b c x - 32c x)\|c --R Type: Union(Expression Integer,...) --E + +--S 101 +t1:=integrate(1/(a*x^2+b*x+c)^(3/2),x) +--R +--R +--------------+ +--R | 2 +-+ +--R - 2x\|a x + b x + c + 2x\|c +--R (2) -------------------------------------------------------- +--R +--------------+ +--R +-+ | 2 2 2 +--R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c +--R Type: Union(Expression Integer,...) +--E + +--S 102 +t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (3) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 103 +bb:=-(a*x^2+b*x+c)/(c^2*x*sqrt(a*x^2+b*x+c))+(b^2-2*a*c)/(2*c^2)*t1-(3*b)/(2*c^2)*t2 +--R +--R (4) +--R +--------------+ +--R 3 2 2 2 | 2 +--R (- 6a b c x - 6b c x - 6b c x)\|a x + b x + c +--R + +--R 2 4 3 3 2 2 2 +-+ +--R (3a b x + (6a b c + 3b )x + 9b c x + 6b c x)\|c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R 3 2 2 2 3 | 2 +--R (2a b c x + 8a c x + 6b c x + 4c )\|a x + b x + c +--R + +--R 2 2 4 3 3 2 2 2 +--R (- 8a c + 2a b )x + (- 12a b c + 2b )x + (- 12a c - 2b c)x +--R + +--R 2 3 +--R - 8b c x - 4c +--R * +--R +-+ +--R \|c +--R / +--R +--------------+ +--R 3 3 3 2 4 +-+ | 2 3 4 +--R (4a c x + 4b c x + 4c x)\|c \|a x + b x + c - 2a b c x +--R + +--R 4 2 3 3 4 2 5 +--R (- 4a c - 2b c )x - 6b c x - 4c x +--R Type: Expression Integer +--E + +--S 104 14:284 Axiom cannot simplify these expressions +cc:=aa-bb +--R +--R (5) +--R 2 4 3 2 3 2 2 2 3 +--R ((72a b c + 6b )x + (144a b c + 108b c)x + 288b c x + 192b c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 3 4 2 2 4 3 +--R (- 48a b c - 36a b c)x + (- 240a b c - 36b c)x +--R + +--R 3 3 2 2 2 3 4 +--R (- 240a b c - 228b c )x - 384b c x - 192b c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R 2 4 3 2 3 2 2 2 +--R (- 72a b c - 6b )x + (- 144a b c - 108b c)x - 288b c x +--R + +--R 3 +--R - 192b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 3 4 2 2 4 3 +--R (48a b c + 36a b c)x + (240a b c + 36b c)x +--R + +--R 3 3 2 2 2 3 4 +--R (240a b c + 228b c )x + 384b c x + 192b c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R 2 4 3 2 3 2 2 2 3 +-+ +--R ((- 60a b c - 5b )x + (- 88a b c - 66b c)x - 112b c x - 32b c )\|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 2 3 4 2 2 4 3 3 3 2 2 +--R (40a b c + 30a b c)x + (152a b c + 26b c)x + (104a b c + 118b c )x +--R + +--R 2 3 4 +--R 128b c x + 32b c +--R / +--R 4 3 3 3 5 2 4 2 5 6 +--R ((48a b c + 4b c )x + (96a c + 72b c )x + 192b c x + 128c ) +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 3 4 4 3 3 3 +--R (- 32a c - 24a b c )x + (- 160a b c - 24b c )x +--R + +--R 5 2 4 2 5 6 +--R (- 160a c - 152b c )x - 256b c x - 128c +--R * +--R +-+ +--R \|c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.295~~~~~$\displaystyle @@ -1160,7 +4765,7 @@ $$ <<*>>= )clear all ---S 16 of 19 +--S 105 14:295 Axiom cannot compute this integral aa:=integrate((a*x^2+b*x+c)^(n+1/2),x) --R --R @@ -1182,7 +4787,7 @@ $$ <<*>>= )clear all ---S 17 of 19 +--S 106 14:296 Axiom cannot compute this integral aa:=integrate(x*(a*x^2+b*x+c)^(n+1/2),x) --R --R @@ -1208,7 +4813,7 @@ $$ <<*>>= )clear all ---S 18 of 19 +--S 107 14:297 Axiom cannot compute this integral aa:=integrate(1/(a*x^2+b*x+c)^(n+1/2),x) --R --R @@ -1239,7 +4844,7 @@ $$ <<*>>= )clear all ---S 19 of 19 +--S 108 14:298 Axiom cannot compute this integral aa:=integrate(1/(x*(a*x^2+b*x+c)^(n+1/2)),x) --R --R