\(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^9} \, dx\) [76]

Optimal result

Integrand size = 22, antiderivative size = 330 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx=\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{8 e^8 (d+e x)^8}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{7 e^8 (d+e x)^7}+\frac {c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{2 e^8 (d+e x)^6}+\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{5 e^8 (d+e x)^5}+\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{4 e^8 (d+e x)^4}-\frac {c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{e^8 (d+e x)^3}+\frac {c^3 (7 B d-A e)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)} \] Output:

1/8*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^8-1/7*(a*e^2+c*d^2)^2*(-6*A*c*d 
*e+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d)^7+1/2*c*(a*e^2+c*d^2)*(-A*a*e^3-5*A*c*d^ 
2*e+3*B*a*d*e^2+7*B*c*d^3)/e^8/(e*x+d)^6+1/5*c*(4*A*c*d*e*(3*a*e^2+5*c*d^2 
)-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d^4))/e^8/(e*x+d)^5+1/4*c^2*(-3*A*a*e 
^3-15*A*c*d^2*e+15*B*a*d*e^2+35*B*c*d^3)/e^8/(e*x+d)^4-c^2*(-2*A*c*d*e+B*a 
*e^2+7*B*c*d^2)/e^8/(e*x+d)^3+1/2*c^3*(-A*e+7*B*d)/e^8/(e*x+d)^2-B*c^3/e^8 
/(e*x+d)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {A e \left (35 a^3 e^6+5 a^2 c e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+3 a c^2 e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )+B \left (5 a^3 e^6 (d+8 e x)+3 a^2 c e^4 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+5 a c^2 e^2 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+35 c^3 \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )\right )}{280 e^8 (d+e x)^8} \] Input:

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^9,x]
 

Output:

-1/280*(A*e*(35*a^3*e^6 + 5*a^2*c*e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*a*c 
^2*e^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5* 
c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 
56*d*e^5*x^5 + 28*e^6*x^6)) + B*(5*a^3*e^6*(d + 8*e*x) + 3*a^2*c*e^4*(d^3 
+ 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 5*a*c^2*e^2*(d^5 + 8*d^4*e*x + 
28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + 35*c^3*(d^7 
 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3*x^3 + 70*d^3*e^4*x^4 + 56*d^2*e 
^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7)))/(e^8*(d + e*x)^8)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 330, 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.

\(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^9} \, dx\)

\(\Big \downarrow \) 652

\(\displaystyle \int \left (-\frac {c \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7 (d+e x)^6}-\frac {3 c^2 \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7 (d+e x)^4}+\frac {c^2 \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7 (d+e x)^5}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^8}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^9}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)^7}+\frac {c^3 (A e-7 B d)}{e^7 (d+e x)^3}+\frac {B c^3}{e^7 (d+e x)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8 (d+e x)^5}-\frac {c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)^3}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{4 e^8 (d+e x)^4}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^7}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{8 e^8 (d+e x)^8}+\frac {c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8 (d+e x)^6}+\frac {c^3 (7 B d-A e)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)}\)

Input:

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^9,x]
 

Output:

((B*d - A*e)*(c*d^2 + a*e^2)^3)/(8*e^8*(d + e*x)^8) - ((c*d^2 + a*e^2)^2*( 
7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(7*e^8*(d + e*x)^7) + (c*(c*d^2 + a*e^2) 
*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(2*e^8*(d + e*x)^6) + 
(c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2 
*e^4)))/(5*e^8*(d + e*x)^5) + (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e 
^2 - 3*a*A*e^3))/(4*e^8*(d + e*x)^4) - (c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e 
^2))/(e^8*(d + e*x)^3) + (c^3*(7*B*d - A*e))/(2*e^8*(d + e*x)^2) - (B*c^3) 
/(e^8*(d + e*x))
 

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 = 0.85 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.28

method	result	size
risch	\(\frac {-\frac {B \,c^{3} x^{7}}{e}-\frac {c^{3} \left (A e +7 B d \right ) x^{6}}{2 e^{2}}-\frac {c^{2} \left (A c d e +B a \,e^{2}+7 B c \,d^{2}\right ) x^{5}}{e^{3}}-\frac {c^{2} \left (3 A a \,e^{3}+5 A c \,d^{2} e +5 B a d \,e^{2}+35 B c \,d^{3}\right ) x^{4}}{4 e^{4}}-\frac {c \left (3 A a c d \,e^{3}+5 A \,c^{2} d^{3} e +3 B \,e^{4} a^{2}+5 B a c \,d^{2} e^{2}+35 B \,c^{2} d^{4}\right ) x^{3}}{5 e^{5}}-\frac {c \left (5 A \,a^{2} e^{5}+3 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e +3 B \,a^{2} d \,e^{4}+5 B a c \,d^{3} e^{2}+35 B \,c^{2} d^{5}\right ) x^{2}}{10 e^{6}}-\frac {\left (5 A \,a^{2} c d \,e^{5}+3 A a \,c^{2} d^{3} e^{3}+5 A \,c^{3} d^{5} e +5 B \,a^{3} e^{6}+3 B \,a^{2} c \,d^{2} e^{4}+5 B a \,c^{2} d^{4} e^{2}+35 B \,c^{3} d^{6}\right ) x}{35 e^{7}}-\frac {35 A \,a^{3} e^{7}+5 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+5 A \,c^{3} d^{6} e +5 B \,a^{3} d \,e^{6}+3 B \,a^{2} c \,d^{3} e^{4}+5 B a \,c^{2} d^{5} e^{2}+35 B \,c^{3} d^{7}}{280 e^{8}}}{\left (e x +d \right )^{8}}\)	\(422\)
norman	\(\frac {-\frac {B \,c^{3} x^{7}}{e}-\frac {\left (A \,c^{3} e +7 B \,c^{3} d \right ) x^{6}}{2 e^{2}}-\frac {\left (A \,c^{3} d e +B \,e^{2} a \,c^{2}+7 B \,c^{3} d^{2}\right ) x^{5}}{e^{3}}-\frac {\left (3 A a \,c^{2} e^{3}+5 A \,c^{3} d^{2} e +5 B a \,c^{2} d \,e^{2}+35 B \,c^{3} d^{3}\right ) x^{4}}{4 e^{4}}-\frac {\left (3 A a \,c^{2} d \,e^{3}+5 A \,c^{3} d^{3} e +3 B \,e^{4} a^{2} c +5 B a \,c^{2} d^{2} e^{2}+35 B \,c^{3} d^{4}\right ) x^{3}}{5 e^{5}}-\frac {\left (5 A \,a^{2} c \,e^{5}+3 A a \,c^{2} d^{2} e^{3}+5 A \,c^{3} d^{4} e +3 B \,a^{2} c d \,e^{4}+5 B a \,c^{2} d^{3} e^{2}+35 B \,c^{3} d^{5}\right ) x^{2}}{10 e^{6}}-\frac {\left (5 A \,a^{2} c d \,e^{5}+3 A a \,c^{2} d^{3} e^{3}+5 A \,c^{3} d^{5} e +5 B \,a^{3} e^{6}+3 B \,a^{2} c \,d^{2} e^{4}+5 B a \,c^{2} d^{4} e^{2}+35 B \,c^{3} d^{6}\right ) x}{35 e^{7}}-\frac {35 A \,a^{3} e^{7}+5 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+5 A \,c^{3} d^{6} e +5 B \,a^{3} d \,e^{6}+3 B \,a^{2} c \,d^{3} e^{4}+5 B a \,c^{2} d^{5} e^{2}+35 B \,c^{3} d^{7}}{280 e^{8}}}{\left (e x +d \right )^{8}}\)	\(445\)
default	\(\frac {c^{2} \left (2 A c d e -B a \,e^{2}-7 B c \,d^{2}\right )}{e^{8} \left (e x +d \right )^{3}}-\frac {c^{2} \left (3 A a \,e^{3}+15 A c \,d^{2} e -15 B a d \,e^{2}-35 B c \,d^{3}\right )}{4 e^{8} \left (e x +d \right )^{4}}-\frac {-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}}{7 e^{8} \left (e x +d \right )^{7}}+\frac {c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{5 e^{8} \left (e x +d \right )^{5}}-\frac {B \,c^{3}}{e^{8} \left (e x +d \right )}-\frac {A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}}{8 e^{8} \left (e x +d \right )^{8}}-\frac {c^{3} \left (A e -7 B d \right )}{2 e^{8} \left (e x +d \right )^{2}}-\frac {c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{2 e^{8} \left (e x +d \right )^{6}}\)	\(448\)
gosper	\(-\frac {280 B \,x^{7} c^{3} e^{7}+140 A \,x^{6} c^{3} e^{7}+980 B \,x^{6} c^{3} d \,e^{6}+280 A \,x^{5} c^{3} d \,e^{6}+280 B \,x^{5} a \,c^{2} e^{7}+1960 B \,x^{5} c^{3} d^{2} e^{5}+210 A \,x^{4} a \,c^{2} e^{7}+350 A \,x^{4} c^{3} d^{2} e^{5}+350 B \,x^{4} a \,c^{2} d \,e^{6}+2450 B \,x^{4} c^{3} d^{3} e^{4}+168 A \,x^{3} a \,c^{2} d \,e^{6}+280 A \,x^{3} c^{3} d^{3} e^{4}+168 B \,x^{3} a^{2} c \,e^{7}+280 B \,x^{3} a \,c^{2} d^{2} e^{5}+1960 B \,x^{3} c^{3} d^{4} e^{3}+140 A \,x^{2} a^{2} c \,e^{7}+84 A \,x^{2} a \,c^{2} d^{2} e^{5}+140 A \,x^{2} c^{3} d^{4} e^{3}+84 B \,x^{2} a^{2} c d \,e^{6}+140 B \,x^{2} a \,c^{2} d^{3} e^{4}+980 B \,x^{2} c^{3} d^{5} e^{2}+40 A x \,a^{2} c d \,e^{6}+24 A x a \,c^{2} d^{3} e^{4}+40 A x \,c^{3} d^{5} e^{2}+40 B x \,a^{3} e^{7}+24 B x \,a^{2} c \,d^{2} e^{5}+40 B x a \,c^{2} d^{4} e^{3}+280 B x \,c^{3} d^{6} e +35 A \,a^{3} e^{7}+5 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+5 A \,c^{3} d^{6} e +5 B \,a^{3} d \,e^{6}+3 B \,a^{2} c \,d^{3} e^{4}+5 B a \,c^{2} d^{5} e^{2}+35 B \,c^{3} d^{7}}{280 \left (e x +d \right )^{8} e^{8}}\)	\(489\)
parallelrisch	\(-\frac {280 B \,x^{7} c^{3} e^{7}+140 A \,x^{6} c^{3} e^{7}+980 B \,x^{6} c^{3} d \,e^{6}+280 A \,x^{5} c^{3} d \,e^{6}+280 B \,x^{5} a \,c^{2} e^{7}+1960 B \,x^{5} c^{3} d^{2} e^{5}+210 A \,x^{4} a \,c^{2} e^{7}+350 A \,x^{4} c^{3} d^{2} e^{5}+350 B \,x^{4} a \,c^{2} d \,e^{6}+2450 B \,x^{4} c^{3} d^{3} e^{4}+168 A \,x^{3} a \,c^{2} d \,e^{6}+280 A \,x^{3} c^{3} d^{3} e^{4}+168 B \,x^{3} a^{2} c \,e^{7}+280 B \,x^{3} a \,c^{2} d^{2} e^{5}+1960 B \,x^{3} c^{3} d^{4} e^{3}+140 A \,x^{2} a^{2} c \,e^{7}+84 A \,x^{2} a \,c^{2} d^{2} e^{5}+140 A \,x^{2} c^{3} d^{4} e^{3}+84 B \,x^{2} a^{2} c d \,e^{6}+140 B \,x^{2} a \,c^{2} d^{3} e^{4}+980 B \,x^{2} c^{3} d^{5} e^{2}+40 A x \,a^{2} c d \,e^{6}+24 A x a \,c^{2} d^{3} e^{4}+40 A x \,c^{3} d^{5} e^{2}+40 B x \,a^{3} e^{7}+24 B x \,a^{2} c \,d^{2} e^{5}+40 B x a \,c^{2} d^{4} e^{3}+280 B x \,c^{3} d^{6} e +35 A \,a^{3} e^{7}+5 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+5 A \,c^{3} d^{6} e +5 B \,a^{3} d \,e^{6}+3 B \,a^{2} c \,d^{3} e^{4}+5 B a \,c^{2} d^{5} e^{2}+35 B \,c^{3} d^{7}}{280 \left (e x +d \right )^{8} e^{8}}\)	\(489\)
orering	\(-\frac {280 B \,x^{7} c^{3} e^{7}+140 A \,x^{6} c^{3} e^{7}+980 B \,x^{6} c^{3} d \,e^{6}+280 A \,x^{5} c^{3} d \,e^{6}+280 B \,x^{5} a \,c^{2} e^{7}+1960 B \,x^{5} c^{3} d^{2} e^{5}+210 A \,x^{4} a \,c^{2} e^{7}+350 A \,x^{4} c^{3} d^{2} e^{5}+350 B \,x^{4} a \,c^{2} d \,e^{6}+2450 B \,x^{4} c^{3} d^{3} e^{4}+168 A \,x^{3} a \,c^{2} d \,e^{6}+280 A \,x^{3} c^{3} d^{3} e^{4}+168 B \,x^{3} a^{2} c \,e^{7}+280 B \,x^{3} a \,c^{2} d^{2} e^{5}+1960 B \,x^{3} c^{3} d^{4} e^{3}+140 A \,x^{2} a^{2} c \,e^{7}+84 A \,x^{2} a \,c^{2} d^{2} e^{5}+140 A \,x^{2} c^{3} d^{4} e^{3}+84 B \,x^{2} a^{2} c d \,e^{6}+140 B \,x^{2} a \,c^{2} d^{3} e^{4}+980 B \,x^{2} c^{3} d^{5} e^{2}+40 A x \,a^{2} c d \,e^{6}+24 A x a \,c^{2} d^{3} e^{4}+40 A x \,c^{3} d^{5} e^{2}+40 B x \,a^{3} e^{7}+24 B x \,a^{2} c \,d^{2} e^{5}+40 B x a \,c^{2} d^{4} e^{3}+280 B x \,c^{3} d^{6} e +35 A \,a^{3} e^{7}+5 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+5 A \,c^{3} d^{6} e +5 B \,a^{3} d \,e^{6}+3 B \,a^{2} c \,d^{3} e^{4}+5 B a \,c^{2} d^{5} e^{2}+35 B \,c^{3} d^{7}}{280 \left (e x +d \right )^{8} e^{8}}\)	\(489\)

Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^9,x,method=_RETURNVERBOSE)
 

Output:

(-B*c^3*x^7/e-1/2*c^3*(A*e+7*B*d)/e^2*x^6-c^2*(A*c*d*e+B*a*e^2+7*B*c*d^2)/ 
e^3*x^5-1/4*c^2*(3*A*a*e^3+5*A*c*d^2*e+5*B*a*d*e^2+35*B*c*d^3)/e^4*x^4-1/5 
*c*(3*A*a*c*d*e^3+5*A*c^2*d^3*e+3*B*a^2*e^4+5*B*a*c*d^2*e^2+35*B*c^2*d^4)/ 
e^5*x^3-1/10*c*(5*A*a^2*e^5+3*A*a*c*d^2*e^3+5*A*c^2*d^4*e+3*B*a^2*d*e^4+5* 
B*a*c*d^3*e^2+35*B*c^2*d^5)/e^6*x^2-1/35*(5*A*a^2*c*d*e^5+3*A*a*c^2*d^3*e^ 
3+5*A*c^3*d^5*e+5*B*a^3*e^6+3*B*a^2*c*d^2*e^4+5*B*a*c^2*d^4*e^2+35*B*c^3*d 
^6)/e^7*x-1/280*(35*A*a^3*e^7+5*A*a^2*c*d^2*e^5+3*A*a*c^2*d^4*e^3+5*A*c^3* 
d^6*e+5*B*a^3*d*e^6+3*B*a^2*c*d^3*e^4+5*B*a*c^2*d^5*e^2+35*B*c^3*d^7)/e^8) 
/(e*x+d)^8
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.61 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {280 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 5 \, A c^{3} d^{6} e + 5 \, B a c^{2} d^{5} e^{2} + 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 5 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7} + 140 \, {\left (7 \, B c^{3} d e^{6} + A c^{3} e^{7}\right )} x^{6} + 280 \, {\left (7 \, B c^{3} d^{2} e^{5} + A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 70 \, {\left (35 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 56 \, {\left (35 \, B c^{3} d^{4} e^{3} + 5 \, A c^{3} d^{3} e^{4} + 5 \, B a c^{2} d^{2} e^{5} + 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 28 \, {\left (35 \, B c^{3} d^{5} e^{2} + 5 \, A c^{3} d^{4} e^{3} + 5 \, B a c^{2} d^{3} e^{4} + 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 5 \, A a^{2} c e^{7}\right )} x^{2} + 8 \, {\left (35 \, B c^{3} d^{6} e + 5 \, A c^{3} d^{5} e^{2} + 5 \, B a c^{2} d^{4} e^{3} + 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 5 \, A a^{2} c d e^{6} + 5 \, B a^{3} e^{7}\right )} x}{280 \, {\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} e^{8}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^9,x, algorithm="fricas")
 

Output:

-1/280*(280*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 5*A*c^3*d^6*e + 5*B*a*c^2*d^5*e 
^2 + 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 + 5*A*a^2*c*d^2*e^5 + 5*B*a^3*d 
*e^6 + 35*A*a^3*e^7 + 140*(7*B*c^3*d*e^6 + A*c^3*e^7)*x^6 + 280*(7*B*c^3*d 
^2*e^5 + A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 70*(35*B*c^3*d^3*e^4 + 5*A*c^3*d 
^2*e^5 + 5*B*a*c^2*d*e^6 + 3*A*a*c^2*e^7)*x^4 + 56*(35*B*c^3*d^4*e^3 + 5*A 
*c^3*d^3*e^4 + 5*B*a*c^2*d^2*e^5 + 3*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 + 
28*(35*B*c^3*d^5*e^2 + 5*A*c^3*d^4*e^3 + 5*B*a*c^2*d^3*e^4 + 3*A*a*c^2*d^2 
*e^5 + 3*B*a^2*c*d*e^6 + 5*A*a^2*c*e^7)*x^2 + 8*(35*B*c^3*d^6*e + 5*A*c^3* 
d^5*e^2 + 5*B*a*c^2*d^4*e^3 + 3*A*a*c^2*d^3*e^4 + 3*B*a^2*c*d^2*e^5 + 5*A* 
a^2*c*d*e^6 + 5*B*a^3*e^7)*x)/(e^16*x^8 + 8*d*e^15*x^7 + 28*d^2*e^14*x^6 + 
 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11*x^3 + 28*d^6*e^10*x^2 + 8 
*d^7*e^9*x + d^8*e^8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**9,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.61 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {280 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 5 \, A c^{3} d^{6} e + 5 \, B a c^{2} d^{5} e^{2} + 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 5 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7} + 140 \, {\left (7 \, B c^{3} d e^{6} + A c^{3} e^{7}\right )} x^{6} + 280 \, {\left (7 \, B c^{3} d^{2} e^{5} + A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 70 \, {\left (35 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 56 \, {\left (35 \, B c^{3} d^{4} e^{3} + 5 \, A c^{3} d^{3} e^{4} + 5 \, B a c^{2} d^{2} e^{5} + 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 28 \, {\left (35 \, B c^{3} d^{5} e^{2} + 5 \, A c^{3} d^{4} e^{3} + 5 \, B a c^{2} d^{3} e^{4} + 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 5 \, A a^{2} c e^{7}\right )} x^{2} + 8 \, {\left (35 \, B c^{3} d^{6} e + 5 \, A c^{3} d^{5} e^{2} + 5 \, B a c^{2} d^{4} e^{3} + 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 5 \, A a^{2} c d e^{6} + 5 \, B a^{3} e^{7}\right )} x}{280 \, {\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} e^{8}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^9,x, algorithm="maxima")
 

Output:

-1/280*(280*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 5*A*c^3*d^6*e + 5*B*a*c^2*d^5*e 
^2 + 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 + 5*A*a^2*c*d^2*e^5 + 5*B*a^3*d 
*e^6 + 35*A*a^3*e^7 + 140*(7*B*c^3*d*e^6 + A*c^3*e^7)*x^6 + 280*(7*B*c^3*d 
^2*e^5 + A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 70*(35*B*c^3*d^3*e^4 + 5*A*c^3*d 
^2*e^5 + 5*B*a*c^2*d*e^6 + 3*A*a*c^2*e^7)*x^4 + 56*(35*B*c^3*d^4*e^3 + 5*A 
*c^3*d^3*e^4 + 5*B*a*c^2*d^2*e^5 + 3*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 + 
28*(35*B*c^3*d^5*e^2 + 5*A*c^3*d^4*e^3 + 5*B*a*c^2*d^3*e^4 + 3*A*a*c^2*d^2 
*e^5 + 3*B*a^2*c*d*e^6 + 5*A*a^2*c*e^7)*x^2 + 8*(35*B*c^3*d^6*e + 5*A*c^3* 
d^5*e^2 + 5*B*a*c^2*d^4*e^3 + 3*A*a*c^2*d^3*e^4 + 3*B*a^2*c*d^2*e^5 + 5*A* 
a^2*c*d*e^6 + 5*B*a^3*e^7)*x)/(e^16*x^8 + 8*d*e^15*x^7 + 28*d^2*e^14*x^6 + 
 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11*x^3 + 28*d^6*e^10*x^2 + 8 
*d^7*e^9*x + d^8*e^8)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {280 \, B c^{3} e^{7} x^{7} + 980 \, B c^{3} d e^{6} x^{6} + 140 \, A c^{3} e^{7} x^{6} + 1960 \, B c^{3} d^{2} e^{5} x^{5} + 280 \, A c^{3} d e^{6} x^{5} + 280 \, B a c^{2} e^{7} x^{5} + 2450 \, B c^{3} d^{3} e^{4} x^{4} + 350 \, A c^{3} d^{2} e^{5} x^{4} + 350 \, B a c^{2} d e^{6} x^{4} + 210 \, A a c^{2} e^{7} x^{4} + 1960 \, B c^{3} d^{4} e^{3} x^{3} + 280 \, A c^{3} d^{3} e^{4} x^{3} + 280 \, B a c^{2} d^{2} e^{5} x^{3} + 168 \, A a c^{2} d e^{6} x^{3} + 168 \, B a^{2} c e^{7} x^{3} + 980 \, B c^{3} d^{5} e^{2} x^{2} + 140 \, A c^{3} d^{4} e^{3} x^{2} + 140 \, B a c^{2} d^{3} e^{4} x^{2} + 84 \, A a c^{2} d^{2} e^{5} x^{2} + 84 \, B a^{2} c d e^{6} x^{2} + 140 \, A a^{2} c e^{7} x^{2} + 280 \, B c^{3} d^{6} e x + 40 \, A c^{3} d^{5} e^{2} x + 40 \, B a c^{2} d^{4} e^{3} x + 24 \, A a c^{2} d^{3} e^{4} x + 24 \, B a^{2} c d^{2} e^{5} x + 40 \, A a^{2} c d e^{6} x + 40 \, B a^{3} e^{7} x + 35 \, B c^{3} d^{7} + 5 \, A c^{3} d^{6} e + 5 \, B a c^{2} d^{5} e^{2} + 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 5 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7}}{280 \, {\left (e x + d\right )}^{8} e^{8}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^9,x, algorithm="giac")
 

Output:

-1/280*(280*B*c^3*e^7*x^7 + 980*B*c^3*d*e^6*x^6 + 140*A*c^3*e^7*x^6 + 1960 
*B*c^3*d^2*e^5*x^5 + 280*A*c^3*d*e^6*x^5 + 280*B*a*c^2*e^7*x^5 + 2450*B*c^ 
3*d^3*e^4*x^4 + 350*A*c^3*d^2*e^5*x^4 + 350*B*a*c^2*d*e^6*x^4 + 210*A*a*c^ 
2*e^7*x^4 + 1960*B*c^3*d^4*e^3*x^3 + 280*A*c^3*d^3*e^4*x^3 + 280*B*a*c^2*d 
^2*e^5*x^3 + 168*A*a*c^2*d*e^6*x^3 + 168*B*a^2*c*e^7*x^3 + 980*B*c^3*d^5*e 
^2*x^2 + 140*A*c^3*d^4*e^3*x^2 + 140*B*a*c^2*d^3*e^4*x^2 + 84*A*a*c^2*d^2* 
e^5*x^2 + 84*B*a^2*c*d*e^6*x^2 + 140*A*a^2*c*e^7*x^2 + 280*B*c^3*d^6*e*x + 
 40*A*c^3*d^5*e^2*x + 40*B*a*c^2*d^4*e^3*x + 24*A*a*c^2*d^3*e^4*x + 24*B*a 
^2*c*d^2*e^5*x + 40*A*a^2*c*d*e^6*x + 40*B*a^3*e^7*x + 35*B*c^3*d^7 + 5*A* 
c^3*d^6*e + 5*B*a*c^2*d^5*e^2 + 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 + 5* 
A*a^2*c*d^2*e^5 + 5*B*a^3*d*e^6 + 35*A*a^3*e^7)/((e*x + d)^8*e^8)
 

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.73 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {5\,B\,a^3\,d\,e^6+40\,B\,a^3\,e^7\,x+35\,A\,a^3\,e^7+3\,B\,a^2\,c\,d^3\,e^4+24\,B\,a^2\,c\,d^2\,e^5\,x+5\,A\,a^2\,c\,d^2\,e^5+84\,B\,a^2\,c\,d\,e^6\,x^2+40\,A\,a^2\,c\,d\,e^6\,x+168\,B\,a^2\,c\,e^7\,x^3+140\,A\,a^2\,c\,e^7\,x^2+5\,B\,a\,c^2\,d^5\,e^2+40\,B\,a\,c^2\,d^4\,e^3\,x+3\,A\,a\,c^2\,d^4\,e^3+140\,B\,a\,c^2\,d^3\,e^4\,x^2+24\,A\,a\,c^2\,d^3\,e^4\,x+280\,B\,a\,c^2\,d^2\,e^5\,x^3+84\,A\,a\,c^2\,d^2\,e^5\,x^2+350\,B\,a\,c^2\,d\,e^6\,x^4+168\,A\,a\,c^2\,d\,e^6\,x^3+280\,B\,a\,c^2\,e^7\,x^5+210\,A\,a\,c^2\,e^7\,x^4+35\,B\,c^3\,d^7+280\,B\,c^3\,d^6\,e\,x+5\,A\,c^3\,d^6\,e+980\,B\,c^3\,d^5\,e^2\,x^2+40\,A\,c^3\,d^5\,e^2\,x+1960\,B\,c^3\,d^4\,e^3\,x^3+140\,A\,c^3\,d^4\,e^3\,x^2+2450\,B\,c^3\,d^3\,e^4\,x^4+280\,A\,c^3\,d^3\,e^4\,x^3+1960\,B\,c^3\,d^2\,e^5\,x^5+350\,A\,c^3\,d^2\,e^5\,x^4+980\,B\,c^3\,d\,e^6\,x^6+280\,A\,c^3\,d\,e^6\,x^5+280\,B\,c^3\,e^7\,x^7+140\,A\,c^3\,e^7\,x^6}{280\,d^8\,e^8+2240\,d^7\,e^9\,x+7840\,d^6\,e^{10}\,x^2+15680\,d^5\,e^{11}\,x^3+19600\,d^4\,e^{12}\,x^4+15680\,d^3\,e^{13}\,x^5+7840\,d^2\,e^{14}\,x^6+2240\,d\,e^{15}\,x^7+280\,e^{16}\,x^8} \] Input:

int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^9,x)
 

Output:

-(35*A*a^3*e^7 + 35*B*c^3*d^7 + 5*B*a^3*d*e^6 + 5*A*c^3*d^6*e + 40*B*a^3*e 
^7*x + 140*A*c^3*e^7*x^6 + 280*B*c^3*e^7*x^7 + 280*B*c^3*d^6*e*x + 3*A*a*c 
^2*d^4*e^3 + 5*A*a^2*c*d^2*e^5 + 5*B*a*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4 + 1 
40*A*a^2*c*e^7*x^2 + 210*A*a*c^2*e^7*x^4 + 168*B*a^2*c*e^7*x^3 + 280*B*a*c 
^2*e^7*x^5 + 40*A*c^3*d^5*e^2*x + 280*A*c^3*d*e^6*x^5 + 980*B*c^3*d*e^6*x^ 
6 + 140*A*c^3*d^4*e^3*x^2 + 280*A*c^3*d^3*e^4*x^3 + 350*A*c^3*d^2*e^5*x^4 
+ 980*B*c^3*d^5*e^2*x^2 + 1960*B*c^3*d^4*e^3*x^3 + 2450*B*c^3*d^3*e^4*x^4 
+ 1960*B*c^3*d^2*e^5*x^5 + 84*A*a*c^2*d^2*e^5*x^2 + 140*B*a*c^2*d^3*e^4*x^ 
2 + 280*B*a*c^2*d^2*e^5*x^3 + 40*A*a^2*c*d*e^6*x + 24*A*a*c^2*d^3*e^4*x + 
168*A*a*c^2*d*e^6*x^3 + 40*B*a*c^2*d^4*e^3*x + 24*B*a^2*c*d^2*e^5*x + 84*B 
*a^2*c*d*e^6*x^2 + 350*B*a*c^2*d*e^6*x^4)/(280*d^8*e^8 + 280*e^16*x^8 + 22 
40*d^7*e^9*x + 2240*d*e^15*x^7 + 7840*d^6*e^10*x^2 + 15680*d^5*e^11*x^3 + 
19600*d^4*e^12*x^4 + 15680*d^3*e^13*x^5 + 7840*d^2*e^14*x^6)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx=\frac {35 b \,c^{3} e^{7} x^{8}-140 a \,c^{3} d \,e^{6} x^{6}-280 a b \,c^{2} d \,e^{6} x^{5}-280 a \,c^{3} d^{2} e^{5} x^{5}-210 a^{2} c^{2} d \,e^{6} x^{4}-350 a b \,c^{2} d^{2} e^{5} x^{4}-350 a \,c^{3} d^{3} e^{4} x^{4}-168 a^{2} b c d \,e^{6} x^{3}-168 a^{2} c^{2} d^{2} e^{5} x^{3}-280 a b \,c^{2} d^{3} e^{4} x^{3}-280 a \,c^{3} d^{4} e^{3} x^{3}-140 a^{3} c d \,e^{6} x^{2}-84 a^{2} b c \,d^{2} e^{5} x^{2}-84 a^{2} c^{2} d^{3} e^{4} x^{2}-140 a b \,c^{2} d^{4} e^{3} x^{2}-140 a \,c^{3} d^{5} e^{2} x^{2}-40 a^{3} b d \,e^{6} x -40 a^{3} c \,d^{2} e^{5} x -24 a^{2} b c \,d^{3} e^{4} x -24 a^{2} c^{2} d^{4} e^{3} x -40 a b \,c^{2} d^{5} e^{2} x -40 a \,c^{3} d^{6} e x -35 a^{4} d \,e^{6}-5 a^{3} b \,d^{2} e^{5}-5 a^{3} c \,d^{3} e^{4}-3 a^{2} b c \,d^{4} e^{3}-3 a^{2} c^{2} d^{5} e^{2}-5 a b \,c^{2} d^{6} e -5 a \,c^{3} d^{7}}{280 d \,e^{7} \left (e^{8} x^{8}+8 d \,e^{7} x^{7}+28 d^{2} e^{6} x^{6}+56 d^{3} e^{5} x^{5}+70 d^{4} e^{4} x^{4}+56 d^{5} e^{3} x^{3}+28 d^{6} e^{2} x^{2}+8 d^{7} e x +d^{8}\right )} \] Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^9,x)
 

Output:

( - 35*a**4*d*e**6 - 5*a**3*b*d**2*e**5 - 40*a**3*b*d*e**6*x - 5*a**3*c*d* 
*3*e**4 - 40*a**3*c*d**2*e**5*x - 140*a**3*c*d*e**6*x**2 - 3*a**2*b*c*d**4 
*e**3 - 24*a**2*b*c*d**3*e**4*x - 84*a**2*b*c*d**2*e**5*x**2 - 168*a**2*b* 
c*d*e**6*x**3 - 3*a**2*c**2*d**5*e**2 - 24*a**2*c**2*d**4*e**3*x - 84*a**2 
*c**2*d**3*e**4*x**2 - 168*a**2*c**2*d**2*e**5*x**3 - 210*a**2*c**2*d*e**6 
*x**4 - 5*a*b*c**2*d**6*e - 40*a*b*c**2*d**5*e**2*x - 140*a*b*c**2*d**4*e* 
*3*x**2 - 280*a*b*c**2*d**3*e**4*x**3 - 350*a*b*c**2*d**2*e**5*x**4 - 280* 
a*b*c**2*d*e**6*x**5 - 5*a*c**3*d**7 - 40*a*c**3*d**6*e*x - 140*a*c**3*d** 
5*e**2*x**2 - 280*a*c**3*d**4*e**3*x**3 - 350*a*c**3*d**3*e**4*x**4 - 280* 
a*c**3*d**2*e**5*x**5 - 140*a*c**3*d*e**6*x**6 + 35*b*c**3*e**7*x**8)/(280 
*d*e**7*(d**8 + 8*d**7*e*x + 28*d**6*e**2*x**2 + 56*d**5*e**3*x**3 + 70*d* 
*4*e**4*x**4 + 56*d**3*e**5*x**5 + 28*d**2*e**6*x**6 + 8*d*e**7*x**7 + e** 
8*x**8))