\(\int \frac {(A+B x) (b x+c x^2)^2}{(d+e x)^8} \, dx\) [21]

Optimal result

Integrand size = 24, antiderivative size = 255 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=\frac {d^2 (B d-A e) (c d-b e)^2}{7 e^6 (d+e x)^7}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{6 e^6 (d+e x)^6}-\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{5 e^6 (d+e x)^5}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{4 e^6 (d+e x)^4}+\frac {c (5 B c d-2 b B e-A c e)}{3 e^6 (d+e x)^3}-\frac {B c^2}{2 e^6 (d+e x)^2} \] Output:

1/7*d^2*(-A*e+B*d)*(-b*e+c*d)^2/e^6/(e*x+d)^7-1/6*d*(-b*e+c*d)*(B*d*(-3*b* 
e+5*c*d)-2*A*e*(-b*e+2*c*d))/e^6/(e*x+d)^6-1/5*(A*e*(b^2*e^2-6*b*c*d*e+6*c 
^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))/e^6/(e*x+d)^5+1/4*(2*A*c*e* 
(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))/e^6/(e*x+d)^4+1/3*c*(-A*c*e 
-2*B*b*e+5*B*c*d)/e^6/(e*x+d)^3-1/2*B*c^2/e^6/(e*x+d)^2
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.02 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {2 A e \left (2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 b c e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (3 b^2 e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+8 b c e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+10 c^2 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{420 e^6 (d+e x)^7} \] Input:

Integrate[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^8,x]
 

Output:

-1/420*(2*A*e*(2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*c*e*(d^3 + 7*d 
^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 2*c^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2* 
x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + B*(3*b^2*e^2*(d^3 + 7*d^2*e*x + 21*d*e 
^2*x^2 + 35*e^3*x^3) + 8*b*c*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^ 
3*x^3 + 35*e^4*x^4) + 10*c^2*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^ 
3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)))/(e^6*(d + e*x)^7)
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 2009}

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.

\(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (\frac {B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-2 A c e (2 c d-b e)}{e^5 (d+e x)^5}+\frac {A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^5 (d+e x)^6}-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^8}+\frac {c (A c e+2 b B e-5 B c d)}{e^5 (d+e x)^4}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^7}+\frac {B c^2}{e^5 (d+e x)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}-\frac {A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac {d^2 (B d-A e) (c d-b e)^2}{7 e^6 (d+e x)^7}+\frac {c (-A c e-2 b B e+5 B c d)}{3 e^6 (d+e x)^3}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{6 e^6 (d+e x)^6}-\frac {B c^2}{2 e^6 (d+e x)^2}\)

Input:

Int[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^8,x]
 

Output:

(d^2*(B*d - A*e)*(c*d - b*e)^2)/(7*e^6*(d + e*x)^7) - (d*(c*d - b*e)*(B*d* 
(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e)))/(6*e^6*(d + e*x)^6) - (A*e*(6*c^2* 
d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))/(5 
*e^6*(d + e*x)^5) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b 
^2*e^2))/(4*e^6*(d + e*x)^4) + (c*(5*B*c*d - 2*b*B*e - A*c*e))/(3*e^6*(d + 
 e*x)^3) - (B*c^2)/(2*e^6*(d + e*x)^2)
 

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.14

method	result	size
risch	\(\frac {-\frac {B \,c^{2} x^{5}}{2 e}-\frac {c \left (2 A c e +4 B b e +5 B c d \right ) x^{4}}{6 e^{2}}-\frac {\left (6 A b c \,e^{2}+4 A \,c^{2} d e +3 B \,e^{2} b^{2}+8 B b c d e +10 B \,c^{2} d^{2}\right ) x^{3}}{12 e^{3}}-\frac {\left (4 A \,b^{2} e^{3}+6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right ) x^{2}}{20 e^{4}}-\frac {d \left (4 A \,b^{2} e^{3}+6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right ) x}{60 e^{5}}-\frac {d^{2} \left (4 A \,b^{2} e^{3}+6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right )}{420 e^{6}}}{\left (e x +d \right )^{7}}\)	\(291\)
default	\(-\frac {c \left (A c e +2 B b e -5 B c d \right )}{3 e^{6} \left (e x +d \right )^{3}}-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +B \,e^{2} b^{2}-8 B b c d e +10 B \,c^{2} d^{2}}{4 e^{6} \left (e x +d \right )^{4}}-\frac {d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{7 e^{6} \left (e x +d \right )^{7}}-\frac {A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}}{5 e^{6} \left (e x +d \right )^{5}}-\frac {B \,c^{2}}{2 e^{6} \left (e x +d \right )^{2}}+\frac {d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right )}{6 e^{6} \left (e x +d \right )^{6}}\)	\(307\)
norman	\(\frac {-\frac {B \,c^{2} x^{5}}{2 e}-\frac {\left (2 A \,c^{2} e^{2}+4 B \,e^{2} b c +5 B \,c^{2} d e \right ) x^{4}}{6 e^{3}}-\frac {\left (6 A b c \,e^{3}+4 A \,c^{2} d \,e^{2}+3 B \,e^{3} b^{2}+8 B b c d \,e^{2}+10 B \,c^{2} d^{2} e \right ) x^{3}}{12 e^{4}}-\frac {\left (4 b^{2} A \,e^{4}+6 A b c d \,e^{3}+4 A \,d^{2} e^{2} c^{2}+3 B \,b^{2} d \,e^{3}+8 B b c \,d^{2} e^{2}+10 B \,c^{2} d^{3} e \right ) x^{2}}{20 e^{5}}-\frac {d \left (4 b^{2} A \,e^{4}+6 A b c d \,e^{3}+4 A \,d^{2} e^{2} c^{2}+3 B \,b^{2} d \,e^{3}+8 B b c \,d^{2} e^{2}+10 B \,c^{2} d^{3} e \right ) x}{60 e^{6}}-\frac {d^{2} \left (4 b^{2} A \,e^{4}+6 A b c d \,e^{3}+4 A \,d^{2} e^{2} c^{2}+3 B \,b^{2} d \,e^{3}+8 B b c \,d^{2} e^{2}+10 B \,c^{2} d^{3} e \right )}{420 e^{7}}}{\left (e x +d \right )^{7}}\)	\(320\)
gosper	\(-\frac {210 B \,x^{5} c^{2} e^{5}+140 A \,x^{4} c^{2} e^{5}+280 B \,x^{4} b c \,e^{5}+350 B \,x^{4} c^{2} d \,e^{4}+210 A \,x^{3} b c \,e^{5}+140 A \,x^{3} c^{2} d \,e^{4}+105 B \,x^{3} b^{2} e^{5}+280 B \,x^{3} b c d \,e^{4}+350 B \,x^{3} c^{2} d^{2} e^{3}+84 A \,x^{2} b^{2} e^{5}+126 A \,x^{2} b c d \,e^{4}+84 A \,x^{2} c^{2} d^{2} e^{3}+63 B \,x^{2} b^{2} d \,e^{4}+168 B \,x^{2} b c \,d^{2} e^{3}+210 B \,x^{2} c^{2} d^{3} e^{2}+28 A x \,b^{2} d \,e^{4}+42 A x b c \,d^{2} e^{3}+28 A x \,c^{2} d^{3} e^{2}+21 B x \,b^{2} d^{2} e^{3}+56 B x b c \,d^{3} e^{2}+70 B x \,c^{2} d^{4} e +4 A \,b^{2} d^{2} e^{3}+6 A b c \,d^{3} e^{2}+4 A \,c^{2} d^{4} e +3 B \,b^{2} d^{3} e^{2}+8 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{420 e^{6} \left (e x +d \right )^{7}}\)	\(341\)
parallelrisch	\(-\frac {210 B \,c^{2} x^{5} e^{6}+140 A \,c^{2} e^{6} x^{4}+280 B b c \,e^{6} x^{4}+350 B \,c^{2} d \,e^{5} x^{4}+210 A b c \,e^{6} x^{3}+140 A \,c^{2} d \,e^{5} x^{3}+105 B \,b^{2} e^{6} x^{3}+280 B b c d \,e^{5} x^{3}+350 B \,c^{2} d^{2} e^{4} x^{3}+84 A \,b^{2} e^{6} x^{2}+126 A b c d \,e^{5} x^{2}+84 A \,c^{2} d^{2} e^{4} x^{2}+63 B \,b^{2} d \,e^{5} x^{2}+168 B b c \,d^{2} e^{4} x^{2}+210 B \,c^{2} d^{3} e^{3} x^{2}+28 A \,b^{2} d \,e^{5} x +42 A b c \,d^{2} e^{4} x +28 A \,c^{2} d^{3} e^{3} x +21 B \,b^{2} d^{2} e^{4} x +56 B b c \,d^{3} e^{3} x +70 B \,c^{2} d^{4} e^{2} x +4 A \,b^{2} d^{2} e^{4}+6 A b c \,d^{3} e^{3}+4 A \,c^{2} d^{4} e^{2}+3 B \,b^{2} d^{3} e^{3}+8 B b c \,d^{4} e^{2}+10 B \,c^{2} d^{5} e}{420 e^{7} \left (e x +d \right )^{7}}\)	\(348\)
orering	\(-\frac {\left (210 B \,x^{5} c^{2} e^{5}+140 A \,x^{4} c^{2} e^{5}+280 B \,x^{4} b c \,e^{5}+350 B \,x^{4} c^{2} d \,e^{4}+210 A \,x^{3} b c \,e^{5}+140 A \,x^{3} c^{2} d \,e^{4}+105 B \,x^{3} b^{2} e^{5}+280 B \,x^{3} b c d \,e^{4}+350 B \,x^{3} c^{2} d^{2} e^{3}+84 A \,x^{2} b^{2} e^{5}+126 A \,x^{2} b c d \,e^{4}+84 A \,x^{2} c^{2} d^{2} e^{3}+63 B \,x^{2} b^{2} d \,e^{4}+168 B \,x^{2} b c \,d^{2} e^{3}+210 B \,x^{2} c^{2} d^{3} e^{2}+28 A x \,b^{2} d \,e^{4}+42 A x b c \,d^{2} e^{3}+28 A x \,c^{2} d^{3} e^{2}+21 B x \,b^{2} d^{2} e^{3}+56 B x b c \,d^{3} e^{2}+70 B x \,c^{2} d^{4} e +4 A \,b^{2} d^{2} e^{3}+6 A b c \,d^{3} e^{2}+4 A \,c^{2} d^{4} e +3 B \,b^{2} d^{3} e^{2}+8 B b c \,d^{4} e +10 B \,c^{2} d^{5}\right ) \left (c \,x^{2}+b x \right )^{2}}{420 e^{6} \left (c x +b \right )^{2} \left (e x +d \right )^{7} x^{2}}\)	\(362\)

Input:

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^8,x,method=_RETURNVERBOSE)
 

Output:

(-1/2*B*c^2*x^5/e-1/6*c/e^2*(2*A*c*e+4*B*b*e+5*B*c*d)*x^4-1/12/e^3*(6*A*b* 
c*e^2+4*A*c^2*d*e+3*B*b^2*e^2+8*B*b*c*d*e+10*B*c^2*d^2)*x^3-1/20/e^4*(4*A* 
b^2*e^3+6*A*b*c*d*e^2+4*A*c^2*d^2*e+3*B*b^2*d*e^2+8*B*b*c*d^2*e+10*B*c^2*d 
^3)*x^2-1/60*d/e^5*(4*A*b^2*e^3+6*A*b*c*d*e^2+4*A*c^2*d^2*e+3*B*b^2*d*e^2+ 
8*B*b*c*d^2*e+10*B*c^2*d^3)*x-1/420*d^2/e^6*(4*A*b^2*e^3+6*A*b*c*d*e^2+4*A 
*c^2*d^2*e+3*B*b^2*d*e^2+8*B*b*c*d^2*e+10*B*c^2*d^3))/(e*x+d)^7
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {210 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 4 \, A b^{2} d^{2} e^{3} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 70 \, {\left (5 \, B c^{2} d e^{4} + 2 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 35 \, {\left (10 \, B c^{2} d^{2} e^{3} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 21 \, {\left (10 \, B c^{2} d^{3} e^{2} + 4 \, A b^{2} e^{5} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 7 \, {\left (10 \, B c^{2} d^{4} e + 4 \, A b^{2} d e^{4} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{420 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="fricas")
 

Output:

-1/420*(210*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 4*A*b^2*d^2*e^3 + 4*(2*B*b*c + 
A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d^3*e^2 + 70*(5*B*c^2*d*e^4 + 2*(2*B*b* 
c + A*c^2)*e^5)*x^4 + 35*(10*B*c^2*d^2*e^3 + 4*(2*B*b*c + A*c^2)*d*e^4 + 3 
*(B*b^2 + 2*A*b*c)*e^5)*x^3 + 21*(10*B*c^2*d^3*e^2 + 4*A*b^2*e^5 + 4*(2*B* 
b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 7*(10*B*c^2*d^4*e 
+ 4*A*b^2*d*e^4 + 4*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^2*e^ 
3)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^ 
4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**8,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {210 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 4 \, A b^{2} d^{2} e^{3} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 70 \, {\left (5 \, B c^{2} d e^{4} + 2 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 35 \, {\left (10 \, B c^{2} d^{2} e^{3} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 21 \, {\left (10 \, B c^{2} d^{3} e^{2} + 4 \, A b^{2} e^{5} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 7 \, {\left (10 \, B c^{2} d^{4} e + 4 \, A b^{2} d e^{4} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{420 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="maxima")
 

Output:

-1/420*(210*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 4*A*b^2*d^2*e^3 + 4*(2*B*b*c + 
A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d^3*e^2 + 70*(5*B*c^2*d*e^4 + 2*(2*B*b* 
c + A*c^2)*e^5)*x^4 + 35*(10*B*c^2*d^2*e^3 + 4*(2*B*b*c + A*c^2)*d*e^4 + 3 
*(B*b^2 + 2*A*b*c)*e^5)*x^3 + 21*(10*B*c^2*d^3*e^2 + 4*A*b^2*e^5 + 4*(2*B* 
b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 7*(10*B*c^2*d^4*e 
+ 4*A*b^2*d*e^4 + 4*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^2*e^ 
3)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^ 
4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {210 \, B c^{2} e^{5} x^{5} + 350 \, B c^{2} d e^{4} x^{4} + 280 \, B b c e^{5} x^{4} + 140 \, A c^{2} e^{5} x^{4} + 350 \, B c^{2} d^{2} e^{3} x^{3} + 280 \, B b c d e^{4} x^{3} + 140 \, A c^{2} d e^{4} x^{3} + 105 \, B b^{2} e^{5} x^{3} + 210 \, A b c e^{5} x^{3} + 210 \, B c^{2} d^{3} e^{2} x^{2} + 168 \, B b c d^{2} e^{3} x^{2} + 84 \, A c^{2} d^{2} e^{3} x^{2} + 63 \, B b^{2} d e^{4} x^{2} + 126 \, A b c d e^{4} x^{2} + 84 \, A b^{2} e^{5} x^{2} + 70 \, B c^{2} d^{4} e x + 56 \, B b c d^{3} e^{2} x + 28 \, A c^{2} d^{3} e^{2} x + 21 \, B b^{2} d^{2} e^{3} x + 42 \, A b c d^{2} e^{3} x + 28 \, A b^{2} d e^{4} x + 10 \, B c^{2} d^{5} + 8 \, B b c d^{4} e + 4 \, A c^{2} d^{4} e + 3 \, B b^{2} d^{3} e^{2} + 6 \, A b c d^{3} e^{2} + 4 \, A b^{2} d^{2} e^{3}}{420 \, {\left (e x + d\right )}^{7} e^{6}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="giac")
 

Output:

-1/420*(210*B*c^2*e^5*x^5 + 350*B*c^2*d*e^4*x^4 + 280*B*b*c*e^5*x^4 + 140* 
A*c^2*e^5*x^4 + 350*B*c^2*d^2*e^3*x^3 + 280*B*b*c*d*e^4*x^3 + 140*A*c^2*d* 
e^4*x^3 + 105*B*b^2*e^5*x^3 + 210*A*b*c*e^5*x^3 + 210*B*c^2*d^3*e^2*x^2 + 
168*B*b*c*d^2*e^3*x^2 + 84*A*c^2*d^2*e^3*x^2 + 63*B*b^2*d*e^4*x^2 + 126*A* 
b*c*d*e^4*x^2 + 84*A*b^2*e^5*x^2 + 70*B*c^2*d^4*e*x + 56*B*b*c*d^3*e^2*x + 
 28*A*c^2*d^3*e^2*x + 21*B*b^2*d^2*e^3*x + 42*A*b*c*d^2*e^3*x + 28*A*b^2*d 
*e^4*x + 10*B*c^2*d^5 + 8*B*b*c*d^4*e + 4*A*c^2*d^4*e + 3*B*b^2*d^3*e^2 + 
6*A*b*c*d^3*e^2 + 4*A*b^2*d^2*e^3)/((e*x + d)^7*e^6)
 

Mupad [B] (verification not implemented)

Time = 10.83 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.40 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\frac {x^3\,\left (3\,B\,b^2\,e^2+8\,B\,b\,c\,d\,e+6\,A\,b\,c\,e^2+10\,B\,c^2\,d^2+4\,A\,c^2\,d\,e\right )}{12\,e^3}+\frac {d^2\,\left (3\,B\,b^2\,d\,e^2+4\,A\,b^2\,e^3+8\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+10\,B\,c^2\,d^3+4\,A\,c^2\,d^2\,e\right )}{420\,e^6}+\frac {x^2\,\left (3\,B\,b^2\,d\,e^2+4\,A\,b^2\,e^3+8\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+10\,B\,c^2\,d^3+4\,A\,c^2\,d^2\,e\right )}{20\,e^4}+\frac {d\,x\,\left (3\,B\,b^2\,d\,e^2+4\,A\,b^2\,e^3+8\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+10\,B\,c^2\,d^3+4\,A\,c^2\,d^2\,e\right )}{60\,e^5}+\frac {c\,x^4\,\left (2\,A\,c\,e+4\,B\,b\,e+5\,B\,c\,d\right )}{6\,e^2}+\frac {B\,c^2\,x^5}{2\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \] Input:

int(((b*x + c*x^2)^2*(A + B*x))/(d + e*x)^8,x)
 

Output:

-((x^3*(3*B*b^2*e^2 + 10*B*c^2*d^2 + 6*A*b*c*e^2 + 4*A*c^2*d*e + 8*B*b*c*d 
*e))/(12*e^3) + (d^2*(4*A*b^2*e^3 + 10*B*c^2*d^3 + 4*A*c^2*d^2*e + 3*B*b^2 
*d*e^2 + 6*A*b*c*d*e^2 + 8*B*b*c*d^2*e))/(420*e^6) + (x^2*(4*A*b^2*e^3 + 1 
0*B*c^2*d^3 + 4*A*c^2*d^2*e + 3*B*b^2*d*e^2 + 6*A*b*c*d*e^2 + 8*B*b*c*d^2* 
e))/(20*e^4) + (d*x*(4*A*b^2*e^3 + 10*B*c^2*d^3 + 4*A*c^2*d^2*e + 3*B*b^2* 
d*e^2 + 6*A*b*c*d*e^2 + 8*B*b*c*d^2*e))/(60*e^5) + (c*x^4*(2*A*c*e + 4*B*b 
*e + 5*B*c*d))/(6*e^2) + (B*c^2*x^5)/(2*e))/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 
 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6 
*e*x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.60 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx=\frac {-210 b \,c^{2} e^{5} x^{5}-140 a \,c^{2} e^{5} x^{4}-280 b^{2} c \,e^{5} x^{4}-350 b \,c^{2} d \,e^{4} x^{4}-210 a b c \,e^{5} x^{3}-140 a \,c^{2} d \,e^{4} x^{3}-105 b^{3} e^{5} x^{3}-280 b^{2} c d \,e^{4} x^{3}-350 b \,c^{2} d^{2} e^{3} x^{3}-84 a \,b^{2} e^{5} x^{2}-126 a b c d \,e^{4} x^{2}-84 a \,c^{2} d^{2} e^{3} x^{2}-63 b^{3} d \,e^{4} x^{2}-168 b^{2} c \,d^{2} e^{3} x^{2}-210 b \,c^{2} d^{3} e^{2} x^{2}-28 a \,b^{2} d \,e^{4} x -42 a b c \,d^{2} e^{3} x -28 a \,c^{2} d^{3} e^{2} x -21 b^{3} d^{2} e^{3} x -56 b^{2} c \,d^{3} e^{2} x -70 b \,c^{2} d^{4} e x -4 a \,b^{2} d^{2} e^{3}-6 a b c \,d^{3} e^{2}-4 a \,c^{2} d^{4} e -3 b^{3} d^{3} e^{2}-8 b^{2} c \,d^{4} e -10 b \,c^{2} d^{5}}{420 e^{6} \left (e^{7} x^{7}+7 d \,e^{6} x^{6}+21 d^{2} e^{5} x^{5}+35 d^{3} e^{4} x^{4}+35 d^{4} e^{3} x^{3}+21 d^{5} e^{2} x^{2}+7 d^{6} e x +d^{7}\right )} \] Input:

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^8,x)
 

Output:

( - 4*a*b**2*d**2*e**3 - 28*a*b**2*d*e**4*x - 84*a*b**2*e**5*x**2 - 6*a*b* 
c*d**3*e**2 - 42*a*b*c*d**2*e**3*x - 126*a*b*c*d*e**4*x**2 - 210*a*b*c*e** 
5*x**3 - 4*a*c**2*d**4*e - 28*a*c**2*d**3*e**2*x - 84*a*c**2*d**2*e**3*x** 
2 - 140*a*c**2*d*e**4*x**3 - 140*a*c**2*e**5*x**4 - 3*b**3*d**3*e**2 - 21* 
b**3*d**2*e**3*x - 63*b**3*d*e**4*x**2 - 105*b**3*e**5*x**3 - 8*b**2*c*d** 
4*e - 56*b**2*c*d**3*e**2*x - 168*b**2*c*d**2*e**3*x**2 - 280*b**2*c*d*e** 
4*x**3 - 280*b**2*c*e**5*x**4 - 10*b*c**2*d**5 - 70*b*c**2*d**4*e*x - 210* 
b*c**2*d**3*e**2*x**2 - 350*b*c**2*d**2*e**3*x**3 - 350*b*c**2*d*e**4*x**4 
 - 210*b*c**2*e**5*x**5)/(420*e**6*(d**7 + 7*d**6*e*x + 21*d**5*e**2*x**2 
+ 35*d**4*e**3*x**3 + 35*d**3*e**4*x**4 + 21*d**2*e**5*x**5 + 7*d*e**6*x** 
6 + e**7*x**7))