\(\int \frac {(A+B x) (d+e x)}{(a+b x+c x^2)^{7/2}} \, dx\) [999]

Optimal result

Integrand size = 25, antiderivative size = 225 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \left (A c (b d-2 a e)-a B (2 c d-b e)+\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {16 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}} \] Output:

1/5*(-2*A*c*(-2*a*e+b*d)+2*a*B*(-b*e+2*c*d)-2*(b^2*B*e-b*c*(A*e+B*d)+2*c*( 
A*c*d-B*a*e))*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)^(5/2)+2/15*(3*b^2*B*e-8*b*c* 
(A*e+B*d)+4*c*(4*A*c*d+B*a*e))*(2*c*x+b)/c/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(3 
/2)-16/15*(3*b^2*B*e-8*b*c*(A*e+B*d)+4*c*(4*A*c*d+B*a*e))*(2*c*x+b)/(-4*a* 
c+b^2)^3/(c*x^2+b*x+a)^(1/2)
 

Mathematica [A] (verified)

Time = 5.20 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.89 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \left (A \left (b^5 (3 d+5 e x)+32 c^2 \left (-3 a^3 e+15 a^2 c d x+20 a c^2 d x^3+8 c^3 d x^5\right )+16 b c^2 \left (15 a^2 (d-e x)-8 c^2 x^4 (-5 d+e x)-20 a c x^2 (-3 d+e x)\right )-16 b^2 c \left (3 a^2 e-15 a c x (d-2 e x)+10 c^2 x^3 (-3 d+2 e x)\right )-40 b^3 c \left (2 c x^2 (-d+3 e x)+a (d+3 e x)\right )+2 b^4 (a e-5 c x (d+4 e x))\right )+B \left (-96 a^3 c (c d-b e)+8 a^2 \left (b^3 e+20 c^3 e x^3-6 b^2 c (d-5 e x)+30 b c^2 x (-d+e x)\right )+b x \left (-128 c^4 d x^4+120 b^2 c^2 x^2 (-2 d+e x)+16 b c^3 x^3 (-20 d+3 e x)+5 b^4 (d+3 e x)+10 b^3 c x (-4 d+9 e x)\right )+2 a \left (32 c^4 e x^5+120 b^2 c^2 x^2 (-2 d+e x)+80 b c^3 x^3 (-2 d+e x)+20 b^3 c x (-3 d+5 e x)+b^4 (d+10 e x)\right )\right )\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \] Input:

Integrate[((A + B*x)*(d + e*x))/(a + b*x + c*x^2)^(7/2),x]
 

Output:

(-2*(A*(b^5*(3*d + 5*e*x) + 32*c^2*(-3*a^3*e + 15*a^2*c*d*x + 20*a*c^2*d*x 
^3 + 8*c^3*d*x^5) + 16*b*c^2*(15*a^2*(d - e*x) - 8*c^2*x^4*(-5*d + e*x) - 
20*a*c*x^2*(-3*d + e*x)) - 16*b^2*c*(3*a^2*e - 15*a*c*x*(d - 2*e*x) + 10*c 
^2*x^3*(-3*d + 2*e*x)) - 40*b^3*c*(2*c*x^2*(-d + 3*e*x) + a*(d + 3*e*x)) + 
 2*b^4*(a*e - 5*c*x*(d + 4*e*x))) + B*(-96*a^3*c*(c*d - b*e) + 8*a^2*(b^3* 
e + 20*c^3*e*x^3 - 6*b^2*c*(d - 5*e*x) + 30*b*c^2*x*(-d + e*x)) + b*x*(-12 
8*c^4*d*x^4 + 120*b^2*c^2*x^2*(-2*d + e*x) + 16*b*c^3*x^3*(-20*d + 3*e*x) 
+ 5*b^4*(d + 3*e*x) + 10*b^3*c*x*(-4*d + 9*e*x)) + 2*a*(32*c^4*e*x^5 + 120 
*b^2*c^2*x^2*(-2*d + e*x) + 80*b*c^3*x^3*(-2*d + e*x) + 20*b^3*c*x*(-3*d + 
 5*e*x) + b^4*(d + 10*e*x)))))/(15*(b^2 - 4*a*c)^3*(a + x*(b + c*x))^(5/2) 
)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1224, 1089, 1088}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1224

\(\displaystyle \frac {2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1089

\(\displaystyle \frac {2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right ) \left (-\frac {8 c \int \frac {1}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right )}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1088

\(\displaystyle \frac {2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\left (\frac {16 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right ) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{5 c \left (b^2-4 a c\right )}\)

Input:

Int[((A + B*x)*(d + e*x))/(a + b*x + c*x^2)^(7/2),x]
 

Output:

(2*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2 
*c*(A*c*d - a*B*e))*x))/(5*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) - ((3* 
b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*((-2*(b + 2*c*x))/(3* 
(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (16*c*(b + 2*c*x))/(3*(b^2 - 4*a* 
c)^2*Sqrt[a + b*x + c*x^2])))/(5*c*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 1088

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 
2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && 
 NeQ[b^2 - 4*a*c, 0]
 

rule 1089

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 
 3)/((p + 1)*(b^2 - 4*a*c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre 
eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1224

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - ( 
b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2)^(p 
+ 1)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c 
*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(c*(p + 1)*(b^2 - 4*a*c))   Int[(a + 
b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, - 
1] &&  !(IntegerQ[p] && NeQ[a, 0] && NiceSqrtQ[b^2 - 4*a*c])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(550\) vs. \(2(215)=430\).

Time = 1.88 (sec) , antiderivative size = 551, normalized size of antiderivative = 2.45

method	result	size
default	\(A d \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )+\left (A e +B d \right ) \left (-\frac {1}{5 c \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {b \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{2 c}\right )+B e \left (-\frac {x}{4 c \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {1}{5 c \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {b \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{2 c}\right )}{8 c}+\frac {a \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{4 c}\right )\)	\(551\)
trager	\(-\frac {2 \left (128 A b \,c^{4} e \,x^{5}-256 A \,c^{5} d \,x^{5}-64 B a \,c^{4} e \,x^{5}-48 B \,b^{2} c^{3} e \,x^{5}+128 B b \,c^{4} d \,x^{5}+320 A \,b^{2} c^{3} e \,x^{4}-640 A b \,c^{4} d \,x^{4}-160 B a b \,c^{3} e \,x^{4}-120 B \,b^{3} c^{2} e \,x^{4}+320 B \,b^{2} c^{3} d \,x^{4}+320 A a b \,c^{3} e \,x^{3}-640 A a \,c^{4} d \,x^{3}+240 A \,b^{3} c^{2} e \,x^{3}-480 A \,b^{2} c^{3} d \,x^{3}-160 B \,a^{2} c^{3} e \,x^{3}-240 B a \,b^{2} c^{2} e \,x^{3}+320 B a b \,c^{3} d \,x^{3}-90 B \,b^{4} c e \,x^{3}+240 B \,b^{3} c^{2} d \,x^{3}+480 A a \,b^{2} c^{2} e \,x^{2}-960 A a b \,c^{3} d \,x^{2}+40 A \,b^{4} c e \,x^{2}-80 A \,b^{3} c^{2} d \,x^{2}-240 B \,a^{2} b \,c^{2} e \,x^{2}-200 B a \,b^{3} c e \,x^{2}+480 B a \,b^{2} c^{2} d \,x^{2}-15 B \,b^{5} e \,x^{2}+40 B \,b^{4} c d \,x^{2}+240 A \,a^{2} b \,c^{2} e x -480 A \,a^{2} c^{3} d x +120 A a \,b^{3} c e x -240 A a \,b^{2} c^{2} d x -5 A \,b^{5} e x +10 A \,b^{4} c d x -240 B \,a^{2} b^{2} c e x +240 B \,a^{2} b \,c^{2} d x -20 B a \,b^{4} e x +120 B a \,b^{3} c d x -5 B \,b^{5} d x +96 A \,a^{3} c^{2} e +48 A \,a^{2} b^{2} c e -240 A \,a^{2} b \,c^{2} d -2 A a \,b^{4} e +40 A a \,b^{3} c d -3 A d \,b^{5}-96 B \,a^{3} b c e +96 B \,a^{3} c^{2} d -8 B e \,a^{2} b^{3}+48 B \,a^{2} b^{2} c d -2 B a \,b^{4} d \right )}{15 \left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}\)	\(586\)
gosper	\(-\frac {2 \left (128 A b \,c^{4} e \,x^{5}-256 A \,c^{5} d \,x^{5}-64 B a \,c^{4} e \,x^{5}-48 B \,b^{2} c^{3} e \,x^{5}+128 B b \,c^{4} d \,x^{5}+320 A \,b^{2} c^{3} e \,x^{4}-640 A b \,c^{4} d \,x^{4}-160 B a b \,c^{3} e \,x^{4}-120 B \,b^{3} c^{2} e \,x^{4}+320 B \,b^{2} c^{3} d \,x^{4}+320 A a b \,c^{3} e \,x^{3}-640 A a \,c^{4} d \,x^{3}+240 A \,b^{3} c^{2} e \,x^{3}-480 A \,b^{2} c^{3} d \,x^{3}-160 B \,a^{2} c^{3} e \,x^{3}-240 B a \,b^{2} c^{2} e \,x^{3}+320 B a b \,c^{3} d \,x^{3}-90 B \,b^{4} c e \,x^{3}+240 B \,b^{3} c^{2} d \,x^{3}+480 A a \,b^{2} c^{2} e \,x^{2}-960 A a b \,c^{3} d \,x^{2}+40 A \,b^{4} c e \,x^{2}-80 A \,b^{3} c^{2} d \,x^{2}-240 B \,a^{2} b \,c^{2} e \,x^{2}-200 B a \,b^{3} c e \,x^{2}+480 B a \,b^{2} c^{2} d \,x^{2}-15 B \,b^{5} e \,x^{2}+40 B \,b^{4} c d \,x^{2}+240 A \,a^{2} b \,c^{2} e x -480 A \,a^{2} c^{3} d x +120 A a \,b^{3} c e x -240 A a \,b^{2} c^{2} d x -5 A \,b^{5} e x +10 A \,b^{4} c d x -240 B \,a^{2} b^{2} c e x +240 B \,a^{2} b \,c^{2} d x -20 B a \,b^{4} e x +120 B a \,b^{3} c d x -5 B \,b^{5} d x +96 A \,a^{3} c^{2} e +48 A \,a^{2} b^{2} c e -240 A \,a^{2} b \,c^{2} d -2 A a \,b^{4} e +40 A a \,b^{3} c d -3 A d \,b^{5}-96 B \,a^{3} b c e +96 B \,a^{3} c^{2} d -8 B e \,a^{2} b^{3}+48 B \,a^{2} b^{2} c d -2 B a \,b^{4} d \right )}{15 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\)	\(608\)
orering	\(-\frac {2 \left (128 A b \,c^{4} e \,x^{5}-256 A \,c^{5} d \,x^{5}-64 B a \,c^{4} e \,x^{5}-48 B \,b^{2} c^{3} e \,x^{5}+128 B b \,c^{4} d \,x^{5}+320 A \,b^{2} c^{3} e \,x^{4}-640 A b \,c^{4} d \,x^{4}-160 B a b \,c^{3} e \,x^{4}-120 B \,b^{3} c^{2} e \,x^{4}+320 B \,b^{2} c^{3} d \,x^{4}+320 A a b \,c^{3} e \,x^{3}-640 A a \,c^{4} d \,x^{3}+240 A \,b^{3} c^{2} e \,x^{3}-480 A \,b^{2} c^{3} d \,x^{3}-160 B \,a^{2} c^{3} e \,x^{3}-240 B a \,b^{2} c^{2} e \,x^{3}+320 B a b \,c^{3} d \,x^{3}-90 B \,b^{4} c e \,x^{3}+240 B \,b^{3} c^{2} d \,x^{3}+480 A a \,b^{2} c^{2} e \,x^{2}-960 A a b \,c^{3} d \,x^{2}+40 A \,b^{4} c e \,x^{2}-80 A \,b^{3} c^{2} d \,x^{2}-240 B \,a^{2} b \,c^{2} e \,x^{2}-200 B a \,b^{3} c e \,x^{2}+480 B a \,b^{2} c^{2} d \,x^{2}-15 B \,b^{5} e \,x^{2}+40 B \,b^{4} c d \,x^{2}+240 A \,a^{2} b \,c^{2} e x -480 A \,a^{2} c^{3} d x +120 A a \,b^{3} c e x -240 A a \,b^{2} c^{2} d x -5 A \,b^{5} e x +10 A \,b^{4} c d x -240 B \,a^{2} b^{2} c e x +240 B \,a^{2} b \,c^{2} d x -20 B a \,b^{4} e x +120 B a \,b^{3} c d x -5 B \,b^{5} d x +96 A \,a^{3} c^{2} e +48 A \,a^{2} b^{2} c e -240 A \,a^{2} b \,c^{2} d -2 A a \,b^{4} e +40 A a \,b^{3} c d -3 A d \,b^{5}-96 B \,a^{3} b c e +96 B \,a^{3} c^{2} d -8 B e \,a^{2} b^{3}+48 B \,a^{2} b^{2} c d -2 B a \,b^{4} d \right )}{15 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\)	\(608\)

Input:

int((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

A*d*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3 
*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/ 
(c*x^2+b*x+a)^(1/2)))+(A*e+B*d)*(-1/5/c/(c*x^2+b*x+a)^(5/2)-1/2*b/c*(2/5*( 
2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c*x+b) 
/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x 
+a)^(1/2))))+B*e*(-1/4*x/c/(c*x^2+b*x+a)^(5/2)-3/8*b/c*(-1/5/c/(c*x^2+b*x+ 
a)^(5/2)-1/2*b/c*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4* 
a*c-b^2)*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2) 
^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+1/4*a/c*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c* 
x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+ 
a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 805 vs. \(2 (214) = 428\).

Time = 82.87 (sec) , antiderivative size = 805, normalized size of antiderivative = 3.58 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")
 

Output:

2/15*(16*(8*(B*b*c^4 - 2*A*c^5)*d - (3*B*b^2*c^3 + 4*(B*a - 2*A*b)*c^4)*e) 
*x^5 + 40*(8*(B*b^2*c^3 - 2*A*b*c^4)*d - (3*B*b^3*c^2 + 4*(B*a*b - 2*A*b^2 
)*c^3)*e)*x^4 + 10*(8*(3*B*b^3*c^2 - 8*A*a*c^4 + 2*(2*B*a*b - 3*A*b^2)*c^3 
)*d - (9*B*b^4*c + 16*(B*a^2 - 2*A*a*b)*c^3 + 24*(B*a*b^2 - A*b^3)*c^2)*e) 
*x^3 + 5*(8*(B*b^4*c - 24*A*a*b*c^3 + 2*(6*B*a*b^2 - A*b^3)*c^2)*d - (3*B* 
b^5 + 48*(B*a^2*b - 2*A*a*b^2)*c^2 + 8*(5*B*a*b^3 - A*b^4)*c)*e)*x^2 - (2* 
B*a*b^4 + 3*A*b^5 - 48*(2*B*a^3 - 5*A*a^2*b)*c^2 - 8*(6*B*a^2*b^2 + 5*A*a* 
b^3)*c)*d - 2*(4*B*a^2*b^3 + A*a*b^4 - 48*A*a^3*c^2 + 24*(2*B*a^3*b - A*a^ 
2*b^2)*c)*e - 5*((B*b^5 + 96*A*a^2*c^3 - 48*(B*a^2*b - A*a*b^2)*c^2 - 2*(1 
2*B*a*b^3 + A*b^4)*c)*d + (4*B*a*b^4 + A*b^5 - 48*A*a^2*b*c^2 + 24*(2*B*a^ 
2*b^2 - A*a*b^3)*c)*e)*x)*sqrt(c*x^2 + b*x + a)/(a^3*b^6 - 12*a^4*b^4*c + 
48*a^5*b^2*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 6 
4*a^3*c^6)*x^6 + 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5 
)*x^5 + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4 
*c^5)*x^4 + (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4* 
b*c^4)*x^3 + 3*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 6 
4*a^5*c^4)*x^2 + 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3 
)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(e*x+d)/(c*x**2+b*x+a)**(7/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 701 vs. \(2 (214) = 428\).

Time = 0.41 (sec) , antiderivative size = 701, normalized size of antiderivative = 3.12 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {2 \, {\left ({\left ({\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (8 \, B b c^{4} d - 16 \, A c^{5} d - 3 \, B b^{2} c^{3} e - 4 \, B a c^{4} e + 8 \, A b c^{4} e\right )} x}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}} + \frac {5 \, {\left (8 \, B b^{2} c^{3} d - 16 \, A b c^{4} d - 3 \, B b^{3} c^{2} e - 4 \, B a b c^{3} e + 8 \, A b^{2} c^{3} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (24 \, B b^{3} c^{2} d + 32 \, B a b c^{3} d - 48 \, A b^{2} c^{3} d - 64 \, A a c^{4} d - 9 \, B b^{4} c e - 24 \, B a b^{2} c^{2} e + 24 \, A b^{3} c^{2} e - 16 \, B a^{2} c^{3} e + 32 \, A a b c^{3} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (8 \, B b^{4} c d + 96 \, B a b^{2} c^{2} d - 16 \, A b^{3} c^{2} d - 192 \, A a b c^{3} d - 3 \, B b^{5} e - 40 \, B a b^{3} c e + 8 \, A b^{4} c e - 48 \, B a^{2} b c^{2} e + 96 \, A a b^{2} c^{2} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {5 \, {\left (B b^{5} d - 24 \, B a b^{3} c d - 2 \, A b^{4} c d - 48 \, B a^{2} b c^{2} d + 48 \, A a b^{2} c^{2} d + 96 \, A a^{2} c^{3} d + 4 \, B a b^{4} e + A b^{5} e + 48 \, B a^{2} b^{2} c e - 24 \, A a b^{3} c e - 48 \, A a^{2} b c^{2} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {2 \, B a b^{4} d + 3 \, A b^{5} d - 48 \, B a^{2} b^{2} c d - 40 \, A a b^{3} c d - 96 \, B a^{3} c^{2} d + 240 \, A a^{2} b c^{2} d + 8 \, B a^{2} b^{3} e + 2 \, A a b^{4} e + 96 \, B a^{3} b c e - 48 \, A a^{2} b^{2} c e - 96 \, A a^{3} c^{2} e}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )}}{15 \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \] Input:

integrate((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")
 

Output:

2/15*(((2*(4*(2*(8*B*b*c^4*d - 16*A*c^5*d - 3*B*b^2*c^3*e - 4*B*a*c^4*e + 
8*A*b*c^4*e)*x/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3) + 5*(8*B*b 
^2*c^3*d - 16*A*b*c^4*d - 3*B*b^3*c^2*e - 4*B*a*b*c^3*e + 8*A*b^2*c^3*e)/( 
b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 5*(24*B*b^3*c^2*d + 3 
2*B*a*b*c^3*d - 48*A*b^2*c^3*d - 64*A*a*c^4*d - 9*B*b^4*c*e - 24*B*a*b^2*c 
^2*e + 24*A*b^3*c^2*e - 16*B*a^2*c^3*e + 32*A*a*b*c^3*e)/(b^6 - 12*a*b^4*c 
 + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 5*(8*B*b^4*c*d + 96*B*a*b^2*c^2*d - 1 
6*A*b^3*c^2*d - 192*A*a*b*c^3*d - 3*B*b^5*e - 40*B*a*b^3*c*e + 8*A*b^4*c*e 
 - 48*B*a^2*b*c^2*e + 96*A*a*b^2*c^2*e)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 
 - 64*a^3*c^3))*x - 5*(B*b^5*d - 24*B*a*b^3*c*d - 2*A*b^4*c*d - 48*B*a^2*b 
*c^2*d + 48*A*a*b^2*c^2*d + 96*A*a^2*c^3*d + 4*B*a*b^4*e + A*b^5*e + 48*B* 
a^2*b^2*c*e - 24*A*a*b^3*c*e - 48*A*a^2*b*c^2*e)/(b^6 - 12*a*b^4*c + 48*a^ 
2*b^2*c^2 - 64*a^3*c^3))*x - (2*B*a*b^4*d + 3*A*b^5*d - 48*B*a^2*b^2*c*d - 
 40*A*a*b^3*c*d - 96*B*a^3*c^2*d + 240*A*a^2*b*c^2*d + 8*B*a^2*b^3*e + 2*A 
*a*b^4*e + 96*B*a^3*b*c*e - 48*A*a^2*b^2*c*e - 96*A*a^3*c^2*e)/(b^6 - 12*a 
*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))/(c*x^2 + b*x + a)^(5/2)
 

Mupad [B] (verification not implemented)

Time = 13.03 (sec) , antiderivative size = 892, normalized size of antiderivative = 3.96 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

int(((A + B*x)*(d + e*x))/(a + b*x + c*x^2)^(7/2),x)
 

Output:

((16*B*c^2*e*x)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (8*B*b*c*e)/(15*(4* 
a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(1/2) - (x*((b*((2*c^2*(( 
2*A*e)/5 + (2*B*d)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e)/(5*(4*a*c^2 - b^2*c 
))))/c - (b*c*((2*A*e)/5 + (2*B*d)/5))/(4*a*c^2 - b^2*c) - (4*A*c^2*d)/(5* 
(4*a*c^2 - b^2*c)) + (4*B*a*c*e)/(5*(4*a*c^2 - b^2*c))) + (a*((2*c^2*((2*A 
*e)/5 + (2*B*d)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e)/(5*(4*a*c^2 - b^2*c))) 
)/c - (2*A*b*c*d)/(5*(4*a*c^2 - b^2*c)))/(a + b*x + c*x^2)^(5/2) + (x*((b* 
((16*c^3*(A*e + B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e) 
/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*c*(32*A*c^2*d + 8*B*b^2*e - 
 20*A*b*c*e + 8*B*a*c*e - 20*B*b*c*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2) 
) - (8*b*c^2*(A*e + B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*B*a*c 
^2*e)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (a*((16*c^3*(A*e + B*d))/(15 
*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e)/(15*(4*a*c^2 - b^2*c)*(4 
*a*c - b^2))))/c + (b*(32*A*c^2*d + 8*B*b^2*e - 20*A*b*c*e + 8*B*a*c*e - 2 
0*B*b*c*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2) 
+ ((b*c*(256*A*c^2*d + 56*B*b^2*e - 128*A*b*c*e + 32*B*a*c*e - 128*B*b*c*d 
))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2) + (2*c^2*x*(256*A*c^2*d + 56*B*b 
^2*e - 128*A*b*c*e + 32*B*a*c*e - 128*B*b*c*d))/(15*(4*a*c^2 - b^2*c)*(4*a 
*c - b^2)^2))/(a + b*x + c*x^2)^(1/2) - ((4*A*c*e - 2*B*b*e + 4*B*c*d)/(15 
*c*(4*a*c - b^2)) + (4*B*e*x)/(15*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/...
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 1492, normalized size of antiderivative = 6.63 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x)
 

Output:

(2*( - 96*sqrt(a + b*x + c*x**2)*a**4*c**2*e + 48*sqrt(a + b*x + c*x**2)*a 
**3*b**2*c*e + 144*sqrt(a + b*x + c*x**2)*a**3*b*c**2*d - 240*sqrt(a + b*x 
 + c*x**2)*a**3*b*c**2*e*x + 480*sqrt(a + b*x + c*x**2)*a**3*c**3*d*x + 10 
*sqrt(a + b*x + c*x**2)*a**2*b**4*e - 88*sqrt(a + b*x + c*x**2)*a**2*b**3* 
c*d + 120*sqrt(a + b*x + c*x**2)*a**2*b**3*c*e*x - 240*sqrt(a + b*x + c*x* 
*2)*a**2*b**2*c**2*e*x**2 + 960*sqrt(a + b*x + c*x**2)*a**2*b*c**3*d*x**2 
- 160*sqrt(a + b*x + c*x**2)*a**2*b*c**3*e*x**3 + 640*sqrt(a + b*x + c*x** 
2)*a**2*c**4*d*x**3 + 5*sqrt(a + b*x + c*x**2)*a*b**5*d + 25*sqrt(a + b*x 
+ c*x**2)*a*b**5*e*x - 130*sqrt(a + b*x + c*x**2)*a*b**4*c*d*x + 160*sqrt( 
a + b*x + c*x**2)*a*b**4*c*e*x**2 - 400*sqrt(a + b*x + c*x**2)*a*b**3*c**2 
*d*x**2 + 160*sqrt(a + b*x + c*x**2)*a*b**2*c**3*d*x**3 - 160*sqrt(a + b*x 
 + c*x**2)*a*b**2*c**3*e*x**4 + 640*sqrt(a + b*x + c*x**2)*a*b*c**4*d*x**4 
 - 64*sqrt(a + b*x + c*x**2)*a*b*c**4*e*x**5 + 256*sqrt(a + b*x + c*x**2)* 
a*c**5*d*x**5 + 5*sqrt(a + b*x + c*x**2)*b**6*d*x + 15*sqrt(a + b*x + c*x* 
*2)*b**6*e*x**2 - 40*sqrt(a + b*x + c*x**2)*b**5*c*d*x**2 + 90*sqrt(a + b* 
x + c*x**2)*b**5*c*e*x**3 - 240*sqrt(a + b*x + c*x**2)*b**4*c**2*d*x**3 + 
120*sqrt(a + b*x + c*x**2)*b**4*c**2*e*x**4 - 320*sqrt(a + b*x + c*x**2)*b 
**3*c**3*d*x**4 + 48*sqrt(a + b*x + c*x**2)*b**3*c**3*e*x**5 - 128*sqrt(a 
+ b*x + c*x**2)*b**2*c**4*d*x**5 + 64*sqrt(c)*a**4*b*c*e - 256*sqrt(c)*a** 
4*c**2*d - 48*sqrt(c)*a**3*b**3*e + 128*sqrt(c)*a**3*b**2*c*d + 192*sqr...