\(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^{11}} \, dx\) [78]

Optimal result

Integrand size = 22, antiderivative size = 334 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11}} \, dx=\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{10 e^8 (d+e x)^{10}}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{9 e^8 (d+e x)^9}+\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{8 e^8 (d+e x)^8}+\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{7 e^8 (d+e x)^7}+\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{6 e^8 (d+e x)^6}-\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{5 e^8 (d+e x)^5}+\frac {c^3 (7 B d-A e)}{4 e^8 (d+e x)^4}-\frac {B c^3}{3 e^8 (d+e x)^3} \] Output:

1/10*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^10-1/9*(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)^9+3/8*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)^8+1/7*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)^7+1/6*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)^6-3/5*c^2*(-2*A*c*d 
*e+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d)^5+1/4*c^3*(-A*e+7*B*d)/e^8/(e*x+d)^4-1/3 
*B*c^3/e^8/(e*x+d)^3
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.07 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {3 A e \left (84 a^3 e^6+7 a^2 c e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+2 a c^2 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+c^3 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )+B \left (28 a^3 e^6 (d+10 e x)+9 a^2 c e^4 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+6 a c^2 e^2 \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+7 c^3 \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )\right )}{2520 e^8 (d+e x)^{10}} \] Input:

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^11,x]
 

Output:

-1/2520*(3*A*e*(84*a^3*e^6 + 7*a^2*c*e^4*(d^2 + 10*d*e*x + 45*e^2*x^2) + 2 
*a*c^2*e^2*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^ 
4) + c^3*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^ 
4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6)) + B*(28*a^3*e^6*(d + 10*e*x) + 9*a^2 
*c*e^4*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 6*a*c^2*e^2*(d^5 
+ 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5* 
x^5) + 7*c^3*(d^7 + 10*d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^ 
3*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6*x^6 + 120*e^7*x^7)))/(e^8*(d + e*x 
)^10)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 334, 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.

\(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^{11}} \, dx\)

\(\Big \downarrow \) 652

\(\displaystyle \int \left (-\frac {c \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7 (d+e x)^8}-\frac {3 c^2 \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7 (d+e x)^6}+\frac {c^2 \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7 (d+e x)^7}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^{10}}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^{11}}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)^9}+\frac {c^3 (A e-7 B d)}{e^7 (d+e x)^5}+\frac {B c^3}{e^7 (d+e x)^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{7 e^8 (d+e x)^7}-\frac {3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{6 e^8 (d+e x)^6}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{9 e^8 (d+e x)^9}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{10 e^8 (d+e x)^{10}}+\frac {3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{8 e^8 (d+e x)^8}+\frac {c^3 (7 B d-A e)}{4 e^8 (d+e x)^4}-\frac {B c^3}{3 e^8 (d+e x)^3}\)

Input:

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^11,x]
 

Output:

((B*d - A*e)*(c*d^2 + a*e^2)^3)/(10*e^8*(d + e*x)^10) - ((c*d^2 + a*e^2)^2 
*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(9*e^8*(d + e*x)^9) + (3*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))/(8*e^8*(d + e*x)^8 
) + (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)))/(7*e^8*(d + e*x)^7) + (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B 
*d*e^2 - 3*a*A*e^3))/(6*e^8*(d + e*x)^6) - (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + 
 a*B*e^2))/(5*e^8*(d + e*x)^5) + (c^3*(7*B*d - A*e))/(4*e^8*(d + e*x)^4) - 
 (B*c^3)/(3*e^8*(d + e*x)^3)
 

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 = 0.88 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.27

method	result	size
risch	\(\frac {-\frac {B \,c^{3} x^{7}}{3 e}-\frac {c^{3} \left (3 A e +7 B d \right ) x^{6}}{12 e^{2}}-\frac {c^{2} \left (3 A c d e +6 B a \,e^{2}+7 B c \,d^{2}\right ) x^{5}}{10 e^{3}}-\frac {c^{2} \left (6 A a \,e^{3}+3 A c \,d^{2} e +6 B a d \,e^{2}+7 B c \,d^{3}\right ) x^{4}}{12 e^{4}}-\frac {c \left (6 A a c d \,e^{3}+3 A \,c^{2} d^{3} e +9 B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+7 B \,c^{2} d^{4}\right ) x^{3}}{21 e^{5}}-\frac {c \left (21 A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+3 A \,c^{2} d^{4} e +9 B \,a^{2} d \,e^{4}+6 B a c \,d^{3} e^{2}+7 B \,c^{2} d^{5}\right ) x^{2}}{56 e^{6}}-\frac {\left (21 A \,a^{2} c d \,e^{5}+6 A a \,c^{2} d^{3} e^{3}+3 A \,c^{3} d^{5} e +28 B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+6 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}\right ) x}{252 e^{7}}-\frac {252 A \,a^{3} e^{7}+21 A \,a^{2} c \,d^{2} e^{5}+6 A a \,c^{2} d^{4} e^{3}+3 A \,c^{3} d^{6} e +28 B \,a^{3} d \,e^{6}+9 B \,a^{2} c \,d^{3} e^{4}+6 B a \,c^{2} d^{5} e^{2}+7 B \,c^{3} d^{7}}{2520 e^{8}}}{\left (e x +d \right )^{10}}\)	\(425\)
default	\(-\frac {B \,c^{3}}{3 e^{8} \left (e x +d \right )^{3}}-\frac {c^{3} \left (A e -7 B d \right )}{4 e^{8} \left (e x +d \right )^{4}}-\frac {-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}}{9 e^{8} \left (e x +d \right )^{9}}+\frac {c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{7 e^{8} \left (e x +d \right )^{7}}+\frac {3 c^{2} \left (2 A c d e -B a \,e^{2}-7 B c \,d^{2}\right )}{5 e^{8} \left (e x +d \right )^{5}}-\frac {3 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{8 e^{8} \left (e x +d \right )^{8}}-\frac {A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}}{10 e^{8} \left (e x +d \right )^{10}}-\frac {c^{2} \left (3 A a \,e^{3}+15 A c \,d^{2} e -15 B a d \,e^{2}-35 B c \,d^{3}\right )}{6 e^{8} \left (e x +d \right )^{6}}\)	\(449\)
norman	\(\frac {-\frac {B \,c^{3} x^{7}}{3 e}-\frac {\left (3 A \,c^{3} e^{3}+7 B \,c^{3} d \,e^{2}\right ) x^{6}}{12 e^{4}}-\frac {\left (3 A \,c^{3} d \,e^{3}+6 B \,e^{4} a \,c^{2}+7 B \,c^{3} d^{2} e^{2}\right ) x^{5}}{10 e^{5}}-\frac {\left (6 A a \,c^{2} e^{5}+3 A \,c^{3} d^{2} e^{3}+6 B a \,c^{2} d \,e^{4}+7 B \,c^{3} d^{3} e^{2}\right ) x^{4}}{12 e^{6}}-\frac {\left (6 A d a \,c^{2} e^{5}+3 A \,c^{3} d^{3} e^{3}+9 B \,e^{6} a^{2} c +6 B a \,c^{2} d^{2} e^{4}+7 B \,c^{3} d^{4} e^{2}\right ) x^{3}}{21 e^{7}}-\frac {\left (21 a^{2} A c \,e^{7}+6 A a \,c^{2} d^{2} e^{5}+3 A \,c^{3} d^{4} e^{3}+9 B \,a^{2} c d \,e^{6}+6 B a \,c^{2} d^{3} e^{4}+7 B \,c^{3} d^{5} e^{2}\right ) x^{2}}{56 e^{8}}-\frac {\left (21 A \,a^{2} c d \,e^{7}+6 A a \,c^{2} d^{3} e^{5}+3 A \,c^{3} d^{5} e^{3}+28 B \,a^{3} e^{8}+9 B \,a^{2} c \,d^{2} e^{6}+6 B a \,c^{2} d^{4} e^{4}+7 B \,c^{3} d^{6} e^{2}\right ) x}{252 e^{9}}-\frac {252 A \,a^{3} e^{9}+21 A \,a^{2} c \,d^{2} e^{7}+6 A a \,c^{2} d^{4} e^{5}+3 A \,c^{3} d^{6} e^{3}+28 B \,a^{3} d \,e^{8}+9 B \,a^{2} c \,d^{3} e^{6}+6 B a \,c^{2} d^{5} e^{4}+7 B \,c^{3} d^{7} e^{2}}{2520 e^{10}}}{\left (e x +d \right )^{10}}\)	\(483\)
gosper	\(-\frac {840 B \,x^{7} c^{3} e^{7}+630 A \,x^{6} c^{3} e^{7}+1470 B \,x^{6} c^{3} d \,e^{6}+756 A \,x^{5} c^{3} d \,e^{6}+1512 B \,x^{5} a \,c^{2} e^{7}+1764 B \,x^{5} c^{3} d^{2} e^{5}+1260 A \,x^{4} a \,c^{2} e^{7}+630 A \,x^{4} c^{3} d^{2} e^{5}+1260 B \,x^{4} a \,c^{2} d \,e^{6}+1470 B \,x^{4} c^{3} d^{3} e^{4}+720 A \,x^{3} a \,c^{2} d \,e^{6}+360 A \,x^{3} c^{3} d^{3} e^{4}+1080 B \,x^{3} a^{2} c \,e^{7}+720 B \,x^{3} a \,c^{2} d^{2} e^{5}+840 B \,x^{3} c^{3} d^{4} e^{3}+945 A \,x^{2} a^{2} c \,e^{7}+270 A \,x^{2} a \,c^{2} d^{2} e^{5}+135 A \,x^{2} c^{3} d^{4} e^{3}+405 B \,x^{2} a^{2} c d \,e^{6}+270 B \,x^{2} a \,c^{2} d^{3} e^{4}+315 B \,x^{2} c^{3} d^{5} e^{2}+210 A x \,a^{2} c d \,e^{6}+60 A x a \,c^{2} d^{3} e^{4}+30 A x \,c^{3} d^{5} e^{2}+280 B x \,a^{3} e^{7}+90 B x \,a^{2} c \,d^{2} e^{5}+60 B x a \,c^{2} d^{4} e^{3}+70 B x \,c^{3} d^{6} e +252 A \,a^{3} e^{7}+21 A \,a^{2} c \,d^{2} e^{5}+6 A a \,c^{2} d^{4} e^{3}+3 A \,c^{3} d^{6} e +28 B \,a^{3} d \,e^{6}+9 B \,a^{2} c \,d^{3} e^{4}+6 B a \,c^{2} d^{5} e^{2}+7 B \,c^{3} d^{7}}{2520 e^{8} \left (e x +d \right )^{10}}\)	\(489\)
orering	\(-\frac {840 B \,x^{7} c^{3} e^{7}+630 A \,x^{6} c^{3} e^{7}+1470 B \,x^{6} c^{3} d \,e^{6}+756 A \,x^{5} c^{3} d \,e^{6}+1512 B \,x^{5} a \,c^{2} e^{7}+1764 B \,x^{5} c^{3} d^{2} e^{5}+1260 A \,x^{4} a \,c^{2} e^{7}+630 A \,x^{4} c^{3} d^{2} e^{5}+1260 B \,x^{4} a \,c^{2} d \,e^{6}+1470 B \,x^{4} c^{3} d^{3} e^{4}+720 A \,x^{3} a \,c^{2} d \,e^{6}+360 A \,x^{3} c^{3} d^{3} e^{4}+1080 B \,x^{3} a^{2} c \,e^{7}+720 B \,x^{3} a \,c^{2} d^{2} e^{5}+840 B \,x^{3} c^{3} d^{4} e^{3}+945 A \,x^{2} a^{2} c \,e^{7}+270 A \,x^{2} a \,c^{2} d^{2} e^{5}+135 A \,x^{2} c^{3} d^{4} e^{3}+405 B \,x^{2} a^{2} c d \,e^{6}+270 B \,x^{2} a \,c^{2} d^{3} e^{4}+315 B \,x^{2} c^{3} d^{5} e^{2}+210 A x \,a^{2} c d \,e^{6}+60 A x a \,c^{2} d^{3} e^{4}+30 A x \,c^{3} d^{5} e^{2}+280 B x \,a^{3} e^{7}+90 B x \,a^{2} c \,d^{2} e^{5}+60 B x a \,c^{2} d^{4} e^{3}+70 B x \,c^{3} d^{6} e +252 A \,a^{3} e^{7}+21 A \,a^{2} c \,d^{2} e^{5}+6 A a \,c^{2} d^{4} e^{3}+3 A \,c^{3} d^{6} e +28 B \,a^{3} d \,e^{6}+9 B \,a^{2} c \,d^{3} e^{4}+6 B a \,c^{2} d^{5} e^{2}+7 B \,c^{3} d^{7}}{2520 e^{8} \left (e x +d \right )^{10}}\)	\(489\)
parallelrisch	\(-\frac {840 B \,c^{3} x^{7} e^{9}+630 A \,c^{3} e^{9} x^{6}+1470 B \,c^{3} d \,e^{8} x^{6}+756 A \,c^{3} d \,e^{8} x^{5}+1512 B a \,c^{2} e^{9} x^{5}+1764 B \,c^{3} d^{2} e^{7} x^{5}+1260 A a \,c^{2} e^{9} x^{4}+630 A \,c^{3} d^{2} e^{7} x^{4}+1260 B a \,c^{2} d \,e^{8} x^{4}+1470 B \,c^{3} d^{3} e^{6} x^{4}+720 A a \,c^{2} d \,e^{8} x^{3}+360 A \,c^{3} d^{3} e^{6} x^{3}+1080 B \,a^{2} c \,e^{9} x^{3}+720 B a \,c^{2} d^{2} e^{7} x^{3}+840 B \,c^{3} d^{4} e^{5} x^{3}+945 A \,a^{2} c \,e^{9} x^{2}+270 A a \,c^{2} d^{2} e^{7} x^{2}+135 A \,c^{3} d^{4} e^{5} x^{2}+405 B \,a^{2} c d \,e^{8} x^{2}+270 B a \,c^{2} d^{3} e^{6} x^{2}+315 B \,c^{3} d^{5} e^{4} x^{2}+210 A \,a^{2} c d \,e^{8} x +60 A a \,c^{2} d^{3} e^{6} x +30 A \,c^{3} d^{5} e^{4} x +280 B \,a^{3} e^{9} x +90 B \,a^{2} c \,d^{2} e^{7} x +60 B a \,c^{2} d^{4} e^{5} x +70 B \,c^{3} d^{6} e^{3} x +252 A \,a^{3} e^{9}+21 A \,a^{2} c \,d^{2} e^{7}+6 A a \,c^{2} d^{4} e^{5}+3 A \,c^{3} d^{6} e^{3}+28 B \,a^{3} d \,e^{8}+9 B \,a^{2} c \,d^{3} e^{6}+6 B a \,c^{2} d^{5} e^{4}+7 B \,c^{3} d^{7} e^{2}}{2520 e^{10} \left (e x +d \right )^{10}}\)	\(496\)

Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^11,x,method=_RETURNVERBOSE)
 

Output:

(-1/3*B*c^3*x^7/e-1/12*c^3/e^2*(3*A*e+7*B*d)*x^6-1/10*c^2/e^3*(3*A*c*d*e+6 
*B*a*e^2+7*B*c*d^2)*x^5-1/12*c^2/e^4*(6*A*a*e^3+3*A*c*d^2*e+6*B*a*d*e^2+7* 
B*c*d^3)*x^4-1/21*c/e^5*(6*A*a*c*d*e^3+3*A*c^2*d^3*e+9*B*a^2*e^4+6*B*a*c*d 
^2*e^2+7*B*c^2*d^4)*x^3-1/56*c/e^6*(21*A*a^2*e^5+6*A*a*c*d^2*e^3+3*A*c^2*d 
^4*e+9*B*a^2*d*e^4+6*B*a*c*d^3*e^2+7*B*c^2*d^5)*x^2-1/252/e^7*(21*A*a^2*c* 
d*e^5+6*A*a*c^2*d^3*e^3+3*A*c^3*d^5*e+28*B*a^3*e^6+9*B*a^2*c*d^2*e^4+6*B*a 
*c^2*d^4*e^2+7*B*c^3*d^6)*x-1/2520/e^8*(252*A*a^3*e^7+21*A*a^2*c*d^2*e^5+6 
*A*a*c^2*d^4*e^3+3*A*c^3*d^6*e+28*B*a^3*d*e^6+9*B*a^2*c*d^3*e^4+6*B*a*c^2* 
d^5*e^2+7*B*c^3*d^7))/(e*x+d)^10
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.67 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {840 \, B c^{3} e^{7} x^{7} + 7 \, B c^{3} d^{7} + 3 \, A c^{3} d^{6} e + 6 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} + 21 \, A a^{2} c d^{2} e^{5} + 28 \, B a^{3} d e^{6} + 252 \, A a^{3} e^{7} + 210 \, {\left (7 \, B c^{3} d e^{6} + 3 \, A c^{3} e^{7}\right )} x^{6} + 252 \, {\left (7 \, B c^{3} d^{2} e^{5} + 3 \, A c^{3} d e^{6} + 6 \, B a c^{2} e^{7}\right )} x^{5} + 210 \, {\left (7 \, B c^{3} d^{3} e^{4} + 3 \, A c^{3} d^{2} e^{5} + 6 \, B a c^{2} d e^{6} + 6 \, A a c^{2} e^{7}\right )} x^{4} + 120 \, {\left (7 \, B c^{3} d^{4} e^{3} + 3 \, A c^{3} d^{3} e^{4} + 6 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 9 \, B a^{2} c e^{7}\right )} x^{3} + 45 \, {\left (7 \, B c^{3} d^{5} e^{2} + 3 \, A c^{3} d^{4} e^{3} + 6 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} + 21 \, A a^{2} c e^{7}\right )} x^{2} + 10 \, {\left (7 \, B c^{3} d^{6} e + 3 \, A c^{3} d^{5} e^{2} + 6 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 21 \, A a^{2} c d e^{6} + 28 \, B a^{3} e^{7}\right )} x}{2520 \, {\left (e^{18} x^{10} + 10 \, d e^{17} x^{9} + 45 \, d^{2} e^{16} x^{8} + 120 \, d^{3} e^{15} x^{7} + 210 \, d^{4} e^{14} x^{6} + 252 \, d^{5} e^{13} x^{5} + 210 \, d^{6} e^{12} x^{4} + 120 \, d^{7} e^{11} x^{3} + 45 \, d^{8} e^{10} x^{2} + 10 \, d^{9} e^{9} x + d^{10} e^{8}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^11,x, algorithm="fricas")
 

Output:

-1/2520*(840*B*c^3*e^7*x^7 + 7*B*c^3*d^7 + 3*A*c^3*d^6*e + 6*B*a*c^2*d^5*e 
^2 + 6*A*a*c^2*d^4*e^3 + 9*B*a^2*c*d^3*e^4 + 21*A*a^2*c*d^2*e^5 + 28*B*a^3 
*d*e^6 + 252*A*a^3*e^7 + 210*(7*B*c^3*d*e^6 + 3*A*c^3*e^7)*x^6 + 252*(7*B* 
c^3*d^2*e^5 + 3*A*c^3*d*e^6 + 6*B*a*c^2*e^7)*x^5 + 210*(7*B*c^3*d^3*e^4 + 
3*A*c^3*d^2*e^5 + 6*B*a*c^2*d*e^6 + 6*A*a*c^2*e^7)*x^4 + 120*(7*B*c^3*d^4* 
e^3 + 3*A*c^3*d^3*e^4 + 6*B*a*c^2*d^2*e^5 + 6*A*a*c^2*d*e^6 + 9*B*a^2*c*e^ 
7)*x^3 + 45*(7*B*c^3*d^5*e^2 + 3*A*c^3*d^4*e^3 + 6*B*a*c^2*d^3*e^4 + 6*A*a 
*c^2*d^2*e^5 + 9*B*a^2*c*d*e^6 + 21*A*a^2*c*e^7)*x^2 + 10*(7*B*c^3*d^6*e + 
 3*A*c^3*d^5*e^2 + 6*B*a*c^2*d^4*e^3 + 6*A*a*c^2*d^3*e^4 + 9*B*a^2*c*d^2*e 
^5 + 21*A*a^2*c*d*e^6 + 28*B*a^3*e^7)*x)/(e^18*x^10 + 10*d*e^17*x^9 + 45*d 
^2*e^16*x^8 + 120*d^3*e^15*x^7 + 210*d^4*e^14*x^6 + 252*d^5*e^13*x^5 + 210 
*d^6*e^12*x^4 + 120*d^7*e^11*x^3 + 45*d^8*e^10*x^2 + 10*d^9*e^9*x + d^10*e 
^8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**11,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.67 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {840 \, B c^{3} e^{7} x^{7} + 7 \, B c^{3} d^{7} + 3 \, A c^{3} d^{6} e + 6 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} + 21 \, A a^{2} c d^{2} e^{5} + 28 \, B a^{3} d e^{6} + 252 \, A a^{3} e^{7} + 210 \, {\left (7 \, B c^{3} d e^{6} + 3 \, A c^{3} e^{7}\right )} x^{6} + 252 \, {\left (7 \, B c^{3} d^{2} e^{5} + 3 \, A c^{3} d e^{6} + 6 \, B a c^{2} e^{7}\right )} x^{5} + 210 \, {\left (7 \, B c^{3} d^{3} e^{4} + 3 \, A c^{3} d^{2} e^{5} + 6 \, B a c^{2} d e^{6} + 6 \, A a c^{2} e^{7}\right )} x^{4} + 120 \, {\left (7 \, B c^{3} d^{4} e^{3} + 3 \, A c^{3} d^{3} e^{4} + 6 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 9 \, B a^{2} c e^{7}\right )} x^{3} + 45 \, {\left (7 \, B c^{3} d^{5} e^{2} + 3 \, A c^{3} d^{4} e^{3} + 6 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} + 21 \, A a^{2} c e^{7}\right )} x^{2} + 10 \, {\left (7 \, B c^{3} d^{6} e + 3 \, A c^{3} d^{5} e^{2} + 6 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 21 \, A a^{2} c d e^{6} + 28 \, B a^{3} e^{7}\right )} x}{2520 \, {\left (e^{18} x^{10} + 10 \, d e^{17} x^{9} + 45 \, d^{2} e^{16} x^{8} + 120 \, d^{3} e^{15} x^{7} + 210 \, d^{4} e^{14} x^{6} + 252 \, d^{5} e^{13} x^{5} + 210 \, d^{6} e^{12} x^{4} + 120 \, d^{7} e^{11} x^{3} + 45 \, d^{8} e^{10} x^{2} + 10 \, d^{9} e^{9} x + d^{10} e^{8}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^11,x, algorithm="maxima")
 

Output:

-1/2520*(840*B*c^3*e^7*x^7 + 7*B*c^3*d^7 + 3*A*c^3*d^6*e + 6*B*a*c^2*d^5*e 
^2 + 6*A*a*c^2*d^4*e^3 + 9*B*a^2*c*d^3*e^4 + 21*A*a^2*c*d^2*e^5 + 28*B*a^3 
*d*e^6 + 252*A*a^3*e^7 + 210*(7*B*c^3*d*e^6 + 3*A*c^3*e^7)*x^6 + 252*(7*B* 
c^3*d^2*e^5 + 3*A*c^3*d*e^6 + 6*B*a*c^2*e^7)*x^5 + 210*(7*B*c^3*d^3*e^4 + 
3*A*c^3*d^2*e^5 + 6*B*a*c^2*d*e^6 + 6*A*a*c^2*e^7)*x^4 + 120*(7*B*c^3*d^4* 
e^3 + 3*A*c^3*d^3*e^4 + 6*B*a*c^2*d^2*e^5 + 6*A*a*c^2*d*e^6 + 9*B*a^2*c*e^ 
7)*x^3 + 45*(7*B*c^3*d^5*e^2 + 3*A*c^3*d^4*e^3 + 6*B*a*c^2*d^3*e^4 + 6*A*a 
*c^2*d^2*e^5 + 9*B*a^2*c*d*e^6 + 21*A*a^2*c*e^7)*x^2 + 10*(7*B*c^3*d^6*e + 
 3*A*c^3*d^5*e^2 + 6*B*a*c^2*d^4*e^3 + 6*A*a*c^2*d^3*e^4 + 9*B*a^2*c*d^2*e 
^5 + 21*A*a^2*c*d*e^6 + 28*B*a^3*e^7)*x)/(e^18*x^10 + 10*d*e^17*x^9 + 45*d 
^2*e^16*x^8 + 120*d^3*e^15*x^7 + 210*d^4*e^14*x^6 + 252*d^5*e^13*x^5 + 210 
*d^6*e^12*x^4 + 120*d^7*e^11*x^3 + 45*d^8*e^10*x^2 + 10*d^9*e^9*x + d^10*e 
^8)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {840 \, B c^{3} e^{7} x^{7} + 1470 \, B c^{3} d e^{6} x^{6} + 630 \, A c^{3} e^{7} x^{6} + 1764 \, B c^{3} d^{2} e^{5} x^{5} + 756 \, A c^{3} d e^{6} x^{5} + 1512 \, B a c^{2} e^{7} x^{5} + 1470 \, B c^{3} d^{3} e^{4} x^{4} + 630 \, A c^{3} d^{2} e^{5} x^{4} + 1260 \, B a c^{2} d e^{6} x^{4} + 1260 \, A a c^{2} e^{7} x^{4} + 840 \, B c^{3} d^{4} e^{3} x^{3} + 360 \, A c^{3} d^{3} e^{4} x^{3} + 720 \, B a c^{2} d^{2} e^{5} x^{3} + 720 \, A a c^{2} d e^{6} x^{3} + 1080 \, B a^{2} c e^{7} x^{3} + 315 \, B c^{3} d^{5} e^{2} x^{2} + 135 \, A c^{3} d^{4} e^{3} x^{2} + 270 \, B a c^{2} d^{3} e^{4} x^{2} + 270 \, A a c^{2} d^{2} e^{5} x^{2} + 405 \, B a^{2} c d e^{6} x^{2} + 945 \, A a^{2} c e^{7} x^{2} + 70 \, B c^{3} d^{6} e x + 30 \, A c^{3} d^{5} e^{2} x + 60 \, B a c^{2} d^{4} e^{3} x + 60 \, A a c^{2} d^{3} e^{4} x + 90 \, B a^{2} c d^{2} e^{5} x + 210 \, A a^{2} c d e^{6} x + 280 \, B a^{3} e^{7} x + 7 \, B c^{3} d^{7} + 3 \, A c^{3} d^{6} e + 6 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} + 21 \, A a^{2} c d^{2} e^{5} + 28 \, B a^{3} d e^{6} + 252 \, A a^{3} e^{7}}{2520 \, {\left (e x + d\right )}^{10} e^{8}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^11,x, algorithm="giac")
 

Output:

-1/2520*(840*B*c^3*e^7*x^7 + 1470*B*c^3*d*e^6*x^6 + 630*A*c^3*e^7*x^6 + 17 
64*B*c^3*d^2*e^5*x^5 + 756*A*c^3*d*e^6*x^5 + 1512*B*a*c^2*e^7*x^5 + 1470*B 
*c^3*d^3*e^4*x^4 + 630*A*c^3*d^2*e^5*x^4 + 1260*B*a*c^2*d*e^6*x^4 + 1260*A 
*a*c^2*e^7*x^4 + 840*B*c^3*d^4*e^3*x^3 + 360*A*c^3*d^3*e^4*x^3 + 720*B*a*c 
^2*d^2*e^5*x^3 + 720*A*a*c^2*d*e^6*x^3 + 1080*B*a^2*c*e^7*x^3 + 315*B*c^3* 
d^5*e^2*x^2 + 135*A*c^3*d^4*e^3*x^2 + 270*B*a*c^2*d^3*e^4*x^2 + 270*A*a*c^ 
2*d^2*e^5*x^2 + 405*B*a^2*c*d*e^6*x^2 + 945*A*a^2*c*e^7*x^2 + 70*B*c^3*d^6 
*e*x + 30*A*c^3*d^5*e^2*x + 60*B*a*c^2*d^4*e^3*x + 60*A*a*c^2*d^3*e^4*x + 
90*B*a^2*c*d^2*e^5*x + 210*A*a^2*c*d*e^6*x + 280*B*a^3*e^7*x + 7*B*c^3*d^7 
 + 3*A*c^3*d^6*e + 6*B*a*c^2*d^5*e^2 + 6*A*a*c^2*d^4*e^3 + 9*B*a^2*c*d^3*e 
^4 + 21*A*a^2*c*d^2*e^5 + 28*B*a^3*d*e^6 + 252*A*a^3*e^7)/((e*x + d)^10*e^ 
8)
 

Mupad [B] (verification not implemented)

Time = 6.24 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.57 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {\frac {28\,B\,a^3\,d\,e^6+252\,A\,a^3\,e^7+9\,B\,a^2\,c\,d^3\,e^4+21\,A\,a^2\,c\,d^2\,e^5+6\,B\,a\,c^2\,d^5\,e^2+6\,A\,a\,c^2\,d^4\,e^3+7\,B\,c^3\,d^7+3\,A\,c^3\,d^6\,e}{2520\,e^8}+\frac {x\,\left (28\,B\,a^3\,e^6+9\,B\,a^2\,c\,d^2\,e^4+21\,A\,a^2\,c\,d\,e^5+6\,B\,a\,c^2\,d^4\,e^2+6\,A\,a\,c^2\,d^3\,e^3+7\,B\,c^3\,d^6+3\,A\,c^3\,d^5\,e\right )}{252\,e^7}+\frac {c^2\,x^4\,\left (7\,B\,c\,d^3+3\,A\,c\,d^2\,e+6\,B\,a\,d\,e^2+6\,A\,a\,e^3\right )}{12\,e^4}+\frac {c\,x^3\,\left (9\,B\,a^2\,e^4+6\,B\,a\,c\,d^2\,e^2+6\,A\,a\,c\,d\,e^3+7\,B\,c^2\,d^4+3\,A\,c^2\,d^3\,e\right )}{21\,e^5}+\frac {c^3\,x^6\,\left (3\,A\,e+7\,B\,d\right )}{12\,e^2}+\frac {c^2\,x^5\,\left (7\,B\,c\,d^2+3\,A\,c\,d\,e+6\,B\,a\,e^2\right )}{10\,e^3}+\frac {c\,x^2\,\left (9\,B\,a^2\,d\,e^4+21\,A\,a^2\,e^5+6\,B\,a\,c\,d^3\,e^2+6\,A\,a\,c\,d^2\,e^3+7\,B\,c^2\,d^5+3\,A\,c^2\,d^4\,e\right )}{56\,e^6}+\frac {B\,c^3\,x^7}{3\,e}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \] Input:

int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^11,x)
 

Output:

-((252*A*a^3*e^7 + 7*B*c^3*d^7 + 28*B*a^3*d*e^6 + 3*A*c^3*d^6*e + 6*A*a*c^ 
2*d^4*e^3 + 21*A*a^2*c*d^2*e^5 + 6*B*a*c^2*d^5*e^2 + 9*B*a^2*c*d^3*e^4)/(2 
520*e^8) + (x*(28*B*a^3*e^6 + 7*B*c^3*d^6 + 3*A*c^3*d^5*e + 6*A*a*c^2*d^3* 
e^3 + 6*B*a*c^2*d^4*e^2 + 9*B*a^2*c*d^2*e^4 + 21*A*a^2*c*d*e^5))/(252*e^7) 
 + (c^2*x^4*(6*A*a*e^3 + 7*B*c*d^3 + 6*B*a*d*e^2 + 3*A*c*d^2*e))/(12*e^4) 
+ (c*x^3*(9*B*a^2*e^4 + 7*B*c^2*d^4 + 3*A*c^2*d^3*e + 6*A*a*c*d*e^3 + 6*B* 
a*c*d^2*e^2))/(21*e^5) + (c^3*x^6*(3*A*e + 7*B*d))/(12*e^2) + (c^2*x^5*(6* 
B*a*e^2 + 7*B*c*d^2 + 3*A*c*d*e))/(10*e^3) + (c*x^2*(21*A*a^2*e^5 + 7*B*c^ 
2*d^5 + 9*B*a^2*d*e^4 + 3*A*c^2*d^4*e + 6*A*a*c*d^2*e^3 + 6*B*a*c*d^3*e^2) 
)/(56*e^6) + (B*c^3*x^7)/(3*e))/(d^10 + e^10*x^10 + 10*d*e^9*x^9 + 45*d^8* 
e^2*x^2 + 120*d^7*e^3*x^3 + 210*d^6*e^4*x^4 + 252*d^5*e^5*x^5 + 210*d^4*e^ 
6*x^6 + 120*d^3*e^7*x^7 + 45*d^2*e^8*x^8 + 10*d^9*e*x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.76 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11}} \, dx=\frac {-840 b \,c^{3} e^{7} x^{7}-630 a \,c^{3} e^{7} x^{6}-1470 b \,c^{3} d \,e^{6} x^{6}-1512 a b \,c^{2} e^{7} x^{5}-756 a \,c^{3} d \,e^{6} x^{5}-1764 b \,c^{3} d^{2} e^{5} x^{5}-1260 a^{2} c^{2} e^{7} x^{4}-1260 a b \,c^{2} d \,e^{6} x^{4}-630 a \,c^{3} d^{2} e^{5} x^{4}-1470 b \,c^{3} d^{3} e^{4} x^{4}-1080 a^{2} b c \,e^{7} x^{3}-720 a^{2} c^{2} d \,e^{6} x^{3}-720 a b \,c^{2} d^{2} e^{5} x^{3}-360 a \,c^{3} d^{3} e^{4} x^{3}-840 b \,c^{3} d^{4} e^{3} x^{3}-945 a^{3} c \,e^{7} x^{2}-405 a^{2} b c d \,e^{6} x^{2}-270 a^{2} c^{2} d^{2} e^{5} x^{2}-270 a b \,c^{2} d^{3} e^{4} x^{2}-135 a \,c^{3} d^{4} e^{3} x^{2}-315 b \,c^{3} d^{5} e^{2} x^{2}-280 a^{3} b \,e^{7} x -210 a^{3} c d \,e^{6} x -90 a^{2} b c \,d^{2} e^{5} x -60 a^{2} c^{2} d^{3} e^{4} x -60 a b \,c^{2} d^{4} e^{3} x -30 a \,c^{3} d^{5} e^{2} x -70 b \,c^{3} d^{6} e x -252 a^{4} e^{7}-28 a^{3} b d \,e^{6}-21 a^{3} c \,d^{2} e^{5}-9 a^{2} b c \,d^{3} e^{4}-6 a^{2} c^{2} d^{4} e^{3}-6 a b \,c^{2} d^{5} e^{2}-3 a \,c^{3} d^{6} e -7 b \,c^{3} d^{7}}{2520 e^{8} \left (e^{10} x^{10}+10 d \,e^{9} x^{9}+45 d^{2} e^{8} x^{8}+120 d^{3} e^{7} x^{7}+210 d^{4} e^{6} x^{6}+252 d^{5} e^{5} x^{5}+210 d^{6} e^{4} x^{4}+120 d^{7} e^{3} x^{3}+45 d^{8} e^{2} x^{2}+10 d^{9} e x +d^{10}\right )} \] Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^11,x)
 

Output:

( - 252*a**4*e**7 - 28*a**3*b*d*e**6 - 280*a**3*b*e**7*x - 21*a**3*c*d**2* 
e**5 - 210*a**3*c*d*e**6*x - 945*a**3*c*e**7*x**2 - 9*a**2*b*c*d**3*e**4 - 
 90*a**2*b*c*d**2*e**5*x - 405*a**2*b*c*d*e**6*x**2 - 1080*a**2*b*c*e**7*x 
**3 - 6*a**2*c**2*d**4*e**3 - 60*a**2*c**2*d**3*e**4*x - 270*a**2*c**2*d** 
2*e**5*x**2 - 720*a**2*c**2*d*e**6*x**3 - 1260*a**2*c**2*e**7*x**4 - 6*a*b 
*c**2*d**5*e**2 - 60*a*b*c**2*d**4*e**3*x - 270*a*b*c**2*d**3*e**4*x**2 - 
720*a*b*c**2*d**2*e**5*x**3 - 1260*a*b*c**2*d*e**6*x**4 - 1512*a*b*c**2*e* 
*7*x**5 - 3*a*c**3*d**6*e - 30*a*c**3*d**5*e**2*x - 135*a*c**3*d**4*e**3*x 
**2 - 360*a*c**3*d**3*e**4*x**3 - 630*a*c**3*d**2*e**5*x**4 - 756*a*c**3*d 
*e**6*x**5 - 630*a*c**3*e**7*x**6 - 7*b*c**3*d**7 - 70*b*c**3*d**6*e*x - 3 
15*b*c**3*d**5*e**2*x**2 - 840*b*c**3*d**4*e**3*x**3 - 1470*b*c**3*d**3*e* 
*4*x**4 - 1764*b*c**3*d**2*e**5*x**5 - 1470*b*c**3*d*e**6*x**6 - 840*b*c** 
3*e**7*x**7)/(2520*e**8*(d**10 + 10*d**9*e*x + 45*d**8*e**2*x**2 + 120*d** 
7*e**3*x**3 + 210*d**6*e**4*x**4 + 252*d**5*e**5*x**5 + 210*d**4*e**6*x**6 
 + 120*d**3*e**7*x**7 + 45*d**2*e**8*x**8 + 10*d*e**9*x**9 + e**10*x**10))