\(\int \frac {(A+B x) (a+c x^2)^2}{(d+e x)^8} \, dx\) [59]

Optimal result

Integrand size = 22, antiderivative size = 206 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx=\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{7 e^6 (d+e x)^7}-\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{6 e^6 (d+e x)^6}+\frac {2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{5 e^6 (d+e x)^5}-\frac {c \left (5 B c d^2-2 A c d e+a B e^2\right )}{2 e^6 (d+e x)^4}+\frac {c^2 (5 B d-A e)}{3 e^6 (d+e x)^3}-\frac {B c^2}{2 e^6 (d+e x)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {2 A e \left (15 a^2 e^4+2 a c e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (5 a^2 e^4 (d+7 e x)+3 a c e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+5 c^2 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{210 e^6 (d+e x)^7} \] Input:

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

Output:

-1/210*(2*A*e*(15*a^2*e^4 + 2*a*c*e^2*(d^2 + 7*d*e*x + 21*e^2*x^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*(5*a^2* 
e^4*(d + 7*e*x) + 3*a*c*e^2*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) 
+ 5*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 = 0.39 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {652, 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.

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

\(\Big \downarrow \) 652

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((B*d - A*e)*(c*d^2 + a*e^2)^2)/(7*e^6*(d + e*x)^7) - ((c*d^2 + a*e^2)*(5* 
B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(6*e^6*(d + e*x)^6) + (2*c*(5*B*c*d^3 - 3* 
A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(5*e^6*(d + e*x)^5) - (c*(5*B*c*d^2 - 
2*A*c*d*e + a*B*e^2))/(2*e^6*(d + e*x)^4) + (c^2*(5*B*d - A*e))/(3*e^6*(d 
+ e*x)^3) - (B*c^2)/(2*e^6*(d + e*x)^2)
 

Defintions of rubi rules used

rule 652

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ 
)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c 
*x^2)^p, x], x] /; FreeQ[{a, 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 = 0.67 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.13

method	result	size
risch	\(\frac {-\frac {B \,c^{2} x^{5}}{2 e}-\frac {c^{2} \left (2 A e +5 B d \right ) x^{4}}{6 e^{2}}-\frac {c \left (2 A c d e +3 B a \,e^{2}+5 B c \,d^{2}\right ) x^{3}}{6 e^{3}}-\frac {c \left (4 A a \,e^{3}+2 A c \,d^{2} e +3 B a d \,e^{2}+5 B c \,d^{3}\right ) x^{2}}{10 e^{4}}-\frac {\left (4 A a c d \,e^{3}+2 A \,c^{2} d^{3} e +5 B \,e^{4} a^{2}+3 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}\right ) x}{30 e^{5}}-\frac {30 A \,a^{2} e^{5}+4 A a c \,d^{2} e^{3}+2 A \,c^{2} d^{4} e +5 B \,a^{2} d \,e^{4}+3 B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{210 e^{6}}}{\left (e x +d \right )^{7}}\)	\(233\)
default	\(-\frac {c^{2} \left (A e -5 B d \right )}{3 e^{6} \left (e x +d \right )^{3}}+\frac {c \left (2 A c d e -B a \,e^{2}-5 B c \,d^{2}\right )}{2 e^{6} \left (e x +d \right )^{4}}-\frac {A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}}{7 e^{6} \left (e x +d \right )^{7}}-\frac {2 c \left (A a \,e^{3}+3 A c \,d^{2} e -3 B a d \,e^{2}-5 B c \,d^{3}\right )}{5 e^{6} \left (e x +d \right )^{5}}-\frac {B \,c^{2}}{2 e^{6} \left (e x +d \right )^{2}}-\frac {-4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}}{6 e^{6} \left (e x +d \right )^{6}}\)	\(249\)
gosper	\(-\frac {105 B \,x^{5} c^{2} e^{5}+70 A \,x^{4} c^{2} e^{5}+175 B \,x^{4} c^{2} d \,e^{4}+70 A \,x^{3} c^{2} d \,e^{4}+105 B \,x^{3} a c \,e^{5}+175 B \,x^{3} c^{2} d^{2} e^{3}+84 A \,x^{2} a c \,e^{5}+42 A \,x^{2} c^{2} d^{2} e^{3}+63 B \,x^{2} a c d \,e^{4}+105 B \,x^{2} c^{2} d^{3} e^{2}+28 A x a c d \,e^{4}+14 A x \,c^{2} d^{3} e^{2}+35 B x \,a^{2} e^{5}+21 B x a c \,d^{2} e^{3}+35 B x \,c^{2} d^{4} e +30 A \,a^{2} e^{5}+4 A a c \,d^{2} e^{3}+2 A \,c^{2} d^{4} e +5 B \,a^{2} d \,e^{4}+3 B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{210 e^{6} \left (e x +d \right )^{7}}\)	\(259\)
orering	\(-\frac {105 B \,x^{5} c^{2} e^{5}+70 A \,x^{4} c^{2} e^{5}+175 B \,x^{4} c^{2} d \,e^{4}+70 A \,x^{3} c^{2} d \,e^{4}+105 B \,x^{3} a c \,e^{5}+175 B \,x^{3} c^{2} d^{2} e^{3}+84 A \,x^{2} a c \,e^{5}+42 A \,x^{2} c^{2} d^{2} e^{3}+63 B \,x^{2} a c d \,e^{4}+105 B \,x^{2} c^{2} d^{3} e^{2}+28 A x a c d \,e^{4}+14 A x \,c^{2} d^{3} e^{2}+35 B x \,a^{2} e^{5}+21 B x a c \,d^{2} e^{3}+35 B x \,c^{2} d^{4} e +30 A \,a^{2} e^{5}+4 A a c \,d^{2} e^{3}+2 A \,c^{2} d^{4} e +5 B \,a^{2} d \,e^{4}+3 B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{210 e^{6} \left (e x +d \right )^{7}}\)	\(259\)
norman	\(\frac {-\frac {B \,c^{2} x^{5}}{2 e}-\frac {\left (2 A \,c^{2} e^{2}+5 B \,c^{2} d e \right ) x^{4}}{6 e^{3}}-\frac {\left (2 A \,c^{2} d \,e^{2}+3 B \,e^{3} a c +5 B \,c^{2} d^{2} e \right ) x^{3}}{6 e^{4}}-\frac {\left (4 A a c \,e^{4}+2 A \,c^{2} d^{2} e^{2}+3 B a c d \,e^{3}+5 B \,c^{2} d^{3} e \right ) x^{2}}{10 e^{5}}-\frac {\left (4 A a c d \,e^{4}+2 A \,c^{2} d^{3} e^{2}+5 B \,e^{5} a^{2}+3 B a c \,d^{2} e^{3}+5 B \,c^{2} d^{4} e \right ) x}{30 e^{6}}-\frac {30 a^{2} A \,e^{6}+4 A a c \,d^{2} e^{4}+2 A \,c^{2} d^{4} e^{2}+5 B \,a^{2} d \,e^{5}+3 B a c \,d^{3} e^{3}+5 B \,c^{2} d^{5} e}{210 e^{7}}}{\left (e x +d \right )^{7}}\)	\(260\)
parallelrisch	\(-\frac {105 B \,c^{2} x^{5} e^{6}+70 A \,c^{2} e^{6} x^{4}+175 B \,c^{2} d \,e^{5} x^{4}+70 A \,c^{2} d \,e^{5} x^{3}+105 B a c \,e^{6} x^{3}+175 B \,c^{2} d^{2} e^{4} x^{3}+84 A a c \,e^{6} x^{2}+42 A \,c^{2} d^{2} e^{4} x^{2}+63 B a c d \,e^{5} x^{2}+105 B \,c^{2} d^{3} e^{3} x^{2}+28 A a c d \,e^{5} x +14 A \,c^{2} d^{3} e^{3} x +35 B \,a^{2} e^{6} x +21 B a c \,d^{2} e^{4} x +35 B \,c^{2} d^{4} e^{2} x +30 a^{2} A \,e^{6}+4 A a c \,d^{2} e^{4}+2 A \,c^{2} d^{4} e^{2}+5 B \,a^{2} d \,e^{5}+3 B a c \,d^{3} e^{3}+5 B \,c^{2} d^{5} e}{210 e^{7} \left (e x +d \right )^{7}}\)	\(264\)

Input:

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

Output:

(-1/2*B*c^2*x^5/e-1/6/e^2*c^2*(2*A*e+5*B*d)*x^4-1/6*c/e^3*(2*A*c*d*e+3*B*a 
*e^2+5*B*c*d^2)*x^3-1/10*c/e^4*(4*A*a*e^3+2*A*c*d^2*e+3*B*a*d*e^2+5*B*c*d^ 
3)*x^2-1/30/e^5*(4*A*a*c*d*e^3+2*A*c^2*d^3*e+5*B*a^2*e^4+3*B*a*c*d^2*e^2+5 
*B*c^2*d^4)*x-1/210/e^6*(30*A*a^2*e^5+4*A*a*c*d^2*e^3+2*A*c^2*d^4*e+5*B*a^ 
2*d*e^4+3*B*a*c*d^3*e^2+5*B*c^2*d^5))/(e*x+d)^7
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {105 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 2 \, A c^{2} d^{4} e + 3 \, B a c d^{3} e^{2} + 4 \, A a c d^{2} e^{3} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5} + 35 \, {\left (5 \, B c^{2} d e^{4} + 2 \, A c^{2} e^{5}\right )} x^{4} + 35 \, {\left (5 \, B c^{2} d^{2} e^{3} + 2 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 21 \, {\left (5 \, B c^{2} d^{3} e^{2} + 2 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + 4 \, A a c e^{5}\right )} x^{2} + 7 \, {\left (5 \, B c^{2} d^{4} e + 2 \, A c^{2} d^{3} e^{2} + 3 \, B a c d^{2} e^{3} + 4 \, A a c d e^{4} + 5 \, B a^{2} e^{5}\right )} x}{210 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \] Input:

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

Output:

-1/210*(105*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 2*A*c^2*d^4*e + 3*B*a*c*d^3*e^2 
+ 4*A*a*c*d^2*e^3 + 5*B*a^2*d*e^4 + 30*A*a^2*e^5 + 35*(5*B*c^2*d*e^4 + 2*A 
*c^2*e^5)*x^4 + 35*(5*B*c^2*d^2*e^3 + 2*A*c^2*d*e^4 + 3*B*a*c*e^5)*x^3 + 2 
1*(5*B*c^2*d^3*e^2 + 2*A*c^2*d^2*e^3 + 3*B*a*c*d*e^4 + 4*A*a*c*e^5)*x^2 + 
7*(5*B*c^2*d^4*e + 2*A*c^2*d^3*e^2 + 3*B*a*c*d^2*e^3 + 4*A*a*c*d*e^4 + 5*B 
*a^2*e^5)*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. \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {105 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 2 \, A c^{2} d^{4} e + 3 \, B a c d^{3} e^{2} + 4 \, A a c d^{2} e^{3} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5} + 35 \, {\left (5 \, B c^{2} d e^{4} + 2 \, A c^{2} e^{5}\right )} x^{4} + 35 \, {\left (5 \, B c^{2} d^{2} e^{3} + 2 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 21 \, {\left (5 \, B c^{2} d^{3} e^{2} + 2 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + 4 \, A a c e^{5}\right )} x^{2} + 7 \, {\left (5 \, B c^{2} d^{4} e + 2 \, A c^{2} d^{3} e^{2} + 3 \, B a c d^{2} e^{3} + 4 \, A a c d e^{4} + 5 \, B a^{2} e^{5}\right )} x}{210 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \] Input:

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

Output:

-1/210*(105*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 2*A*c^2*d^4*e + 3*B*a*c*d^3*e^2 
+ 4*A*a*c*d^2*e^3 + 5*B*a^2*d*e^4 + 30*A*a^2*e^5 + 35*(5*B*c^2*d*e^4 + 2*A 
*c^2*e^5)*x^4 + 35*(5*B*c^2*d^2*e^3 + 2*A*c^2*d*e^4 + 3*B*a*c*e^5)*x^3 + 2 
1*(5*B*c^2*d^3*e^2 + 2*A*c^2*d^2*e^3 + 3*B*a*c*d*e^4 + 4*A*a*c*e^5)*x^2 + 
7*(5*B*c^2*d^4*e + 2*A*c^2*d^3*e^2 + 3*B*a*c*d^2*e^3 + 4*A*a*c*d*e^4 + 5*B 
*a^2*e^5)*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 = 0.12 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.25 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {105 \, B c^{2} e^{5} x^{5} + 175 \, B c^{2} d e^{4} x^{4} + 70 \, A c^{2} e^{5} x^{4} + 175 \, B c^{2} d^{2} e^{3} x^{3} + 70 \, A c^{2} d e^{4} x^{3} + 105 \, B a c e^{5} x^{3} + 105 \, B c^{2} d^{3} e^{2} x^{2} + 42 \, A c^{2} d^{2} e^{3} x^{2} + 63 \, B a c d e^{4} x^{2} + 84 \, A a c e^{5} x^{2} + 35 \, B c^{2} d^{4} e x + 14 \, A c^{2} d^{3} e^{2} x + 21 \, B a c d^{2} e^{3} x + 28 \, A a c d e^{4} x + 35 \, B a^{2} e^{5} x + 5 \, B c^{2} d^{5} + 2 \, A c^{2} d^{4} e + 3 \, B a c d^{3} e^{2} + 4 \, A a c d^{2} e^{3} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5}}{210 \, {\left (e x + d\right )}^{7} e^{6}} \] Input:

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

Output:

-1/210*(105*B*c^2*e^5*x^5 + 175*B*c^2*d*e^4*x^4 + 70*A*c^2*e^5*x^4 + 175*B 
*c^2*d^2*e^3*x^3 + 70*A*c^2*d*e^4*x^3 + 105*B*a*c*e^5*x^3 + 105*B*c^2*d^3* 
e^2*x^2 + 42*A*c^2*d^2*e^3*x^2 + 63*B*a*c*d*e^4*x^2 + 84*A*a*c*e^5*x^2 + 3 
5*B*c^2*d^4*e*x + 14*A*c^2*d^3*e^2*x + 21*B*a*c*d^2*e^3*x + 28*A*a*c*d*e^4 
*x + 35*B*a^2*e^5*x + 5*B*c^2*d^5 + 2*A*c^2*d^4*e + 3*B*a*c*d^3*e^2 + 4*A* 
a*c*d^2*e^3 + 5*B*a^2*d*e^4 + 30*A*a^2*e^5)/((e*x + d)^7*e^6)
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.45 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\frac {5\,B\,a^2\,d\,e^4+30\,A\,a^2\,e^5+3\,B\,a\,c\,d^3\,e^2+4\,A\,a\,c\,d^2\,e^3+5\,B\,c^2\,d^5+2\,A\,c^2\,d^4\,e}{210\,e^6}+\frac {x\,\left (5\,B\,a^2\,e^4+3\,B\,a\,c\,d^2\,e^2+4\,A\,a\,c\,d\,e^3+5\,B\,c^2\,d^4+2\,A\,c^2\,d^3\,e\right )}{30\,e^5}+\frac {c\,x^3\,\left (5\,B\,c\,d^2+2\,A\,c\,d\,e+3\,B\,a\,e^2\right )}{6\,e^3}+\frac {c^2\,x^4\,\left (2\,A\,e+5\,B\,d\right )}{6\,e^2}+\frac {c\,x^2\,\left (5\,B\,c\,d^3+2\,A\,c\,d^2\,e+3\,B\,a\,d\,e^2+4\,A\,a\,e^3\right )}{10\,e^4}+\frac {B\,c^2\,x^5}{2\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \] Input:

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

Output:

-((30*A*a^2*e^5 + 5*B*c^2*d^5 + 5*B*a^2*d*e^4 + 2*A*c^2*d^4*e + 4*A*a*c*d^ 
2*e^3 + 3*B*a*c*d^3*e^2)/(210*e^6) + (x*(5*B*a^2*e^4 + 5*B*c^2*d^4 + 2*A*c 
^2*d^3*e + 4*A*a*c*d*e^3 + 3*B*a*c*d^2*e^2))/(30*e^5) + (c*x^3*(3*B*a*e^2 
+ 5*B*c*d^2 + 2*A*c*d*e))/(6*e^3) + (c^2*x^4*(2*A*e + 5*B*d))/(6*e^2) + (c 
*x^2*(4*A*a*e^3 + 5*B*c*d^3 + 3*B*a*d*e^2 + 2*A*c*d^2*e))/(10*e^4) + (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 = 0.24 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.58 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx=\frac {-105 b \,c^{2} e^{5} x^{5}-70 a \,c^{2} e^{5} x^{4}-175 b \,c^{2} d \,e^{4} x^{4}-105 a b c \,e^{5} x^{3}-70 a \,c^{2} d \,e^{4} x^{3}-175 b \,c^{2} d^{2} e^{3} x^{3}-84 a^{2} c \,e^{5} x^{2}-63 a b c d \,e^{4} x^{2}-42 a \,c^{2} d^{2} e^{3} x^{2}-105 b \,c^{2} d^{3} e^{2} x^{2}-35 a^{2} b \,e^{5} x -28 a^{2} c d \,e^{4} x -21 a b c \,d^{2} e^{3} x -14 a \,c^{2} d^{3} e^{2} x -35 b \,c^{2} d^{4} e x -30 a^{3} e^{5}-5 a^{2} b d \,e^{4}-4 a^{2} c \,d^{2} e^{3}-3 a b c \,d^{3} e^{2}-2 a \,c^{2} d^{4} e -5 b \,c^{2} d^{5}}{210 e^{6} \left (e^{7} x^{7}+7 d \,e^{6} x^{6}+21 d^{2} e^{5} x^{5}+35 d^{3} e^{4} x^{4}+35 d^{4} e^{3} x^{3}+21 d^{5} e^{2} x^{2}+7 d^{6} e x +d^{7}\right )} \] Input:

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

Output:

( - 30*a**3*e**5 - 5*a**2*b*d*e**4 - 35*a**2*b*e**5*x - 4*a**2*c*d**2*e**3 
 - 28*a**2*c*d*e**4*x - 84*a**2*c*e**5*x**2 - 3*a*b*c*d**3*e**2 - 21*a*b*c 
*d**2*e**3*x - 63*a*b*c*d*e**4*x**2 - 105*a*b*c*e**5*x**3 - 2*a*c**2*d**4* 
e - 14*a*c**2*d**3*e**2*x - 42*a*c**2*d**2*e**3*x**2 - 70*a*c**2*d*e**4*x* 
*3 - 70*a*c**2*e**5*x**4 - 5*b*c**2*d**5 - 35*b*c**2*d**4*e*x - 105*b*c**2 
*d**3*e**2*x**2 - 175*b*c**2*d**2*e**3*x**3 - 175*b*c**2*d*e**4*x**4 - 105 
*b*c**2*e**5*x**5)/(210*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))