\(\int (A+B x) (d+e x)^3 (a+c x^2)^3 \, dx\) [64]

Optimal result

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

-1/4*(-A*e+B*d)*(a*e^2+c*d^2)^3*(e*x+d)^4/e^8+1/5*(a*e^2+c*d^2)^2*(-6*A*c* 
d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^5/e^8-1/2*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*x+d)^6/e^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*x+d)^7/e^8-1/8*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*x+d)^8/e^8+1/3*c^2*(-2*A*c*d* 
e+B*a*e^2+7*B*c*d^2)*(e*x+d)^9/e^8-1/10*c^3*(-A*e+7*B*d)*(e*x+d)^10/e^8+1/ 
11*B*c^3*(e*x+d)^11/e^8
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.97 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^3 \, dx=a^3 A d^3 x+\frac {1}{2} a^3 d^2 (B d+3 A e) x^2+a^2 d \left (A c d^2+a B d e+a A e^2\right ) x^3+\frac {1}{4} a^2 \left (3 B c d^3+9 A c d^2 e+3 a B d e^2+a A e^3\right ) x^4+\frac {1}{5} a \left (a B e \left (9 c d^2+a e^2\right )+3 A c d \left (c d^2+3 a e^2\right )\right ) x^5+\frac {1}{2} a c \left (B c d^3+3 A c d^2 e+3 a B d e^2+a A e^3\right ) x^6+\frac {1}{7} c \left (3 a B e \left (3 c d^2+a e^2\right )+A c d \left (c d^2+9 a e^2\right )\right ) x^7+\frac {1}{8} c^2 \left (B c d^3+3 A c d^2 e+9 a B d e^2+3 a A e^3\right ) x^8+\frac {1}{3} c^2 e \left (B c d^2+A c d e+a B e^2\right ) x^9+\frac {1}{10} c^3 e^2 (3 B d+A e) x^{10}+\frac {1}{11} B c^3 e^3 x^{11} \] Input:

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

Output:

a^3*A*d^3*x + (a^3*d^2*(B*d + 3*A*e)*x^2)/2 + a^2*d*(A*c*d^2 + a*B*d*e + a 
*A*e^2)*x^3 + (a^2*(3*B*c*d^3 + 9*A*c*d^2*e + 3*a*B*d*e^2 + a*A*e^3)*x^4)/ 
4 + (a*(a*B*e*(9*c*d^2 + a*e^2) + 3*A*c*d*(c*d^2 + 3*a*e^2))*x^5)/5 + (a*c 
*(B*c*d^3 + 3*A*c*d^2*e + 3*a*B*d*e^2 + a*A*e^3)*x^6)/2 + (c*(3*a*B*e*(3*c 
*d^2 + a*e^2) + A*c*d*(c*d^2 + 9*a*e^2))*x^7)/7 + (c^2*(B*c*d^3 + 3*A*c*d^ 
2*e + 9*a*B*d*e^2 + 3*a*A*e^3)*x^8)/8 + (c^2*e*(B*c*d^2 + A*c*d*e + a*B*e^ 
2)*x^9)/3 + (c^3*e^2*(3*B*d + A*e)*x^10)/10 + (B*c^3*e^3*x^11)/11
 

Rubi [A] (verified)

Time = 0.68 (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.

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

\(\Big \downarrow \) 652

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/4*((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^4)/e^8 + ((c*d^2 + a*e^2)^2* 
(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d + e*x)^5)/(5*e^8) - (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)*(d + e*x)^6)/(2*e^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))*(d + e*x)^7)/(7*e^8) - (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d* 
e^2 - 3*a*A*e^3)*(d + e*x)^8)/(8*e^8) + (c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B* 
e^2)*(d + e*x)^9)/(3*e^8) - (c^3*(7*B*d - A*e)*(d + e*x)^10)/(10*e^8) + (B 
*c^3*(d + e*x)^11)/(11*e^8)
 

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 = 0.58 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {B \,e^{3} c^{3} x^{11}}{11}+\frac {\left (A \,e^{3}+3 B d \,e^{2}\right ) c^{3} x^{10}}{10}+\frac {\left (\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c^{3}+3 B \,e^{3} a \,c^{2}\right ) x^{9}}{9}+\frac {\left (\left (3 A \,d^{2} e +B \,d^{3}\right ) c^{3}+3 \left (A \,e^{3}+3 B d \,e^{2}\right ) a \,c^{2}\right ) x^{8}}{8}+\frac {\left (A \,c^{3} d^{3}+3 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a \,c^{2}+3 B \,e^{3} a^{2} c \right ) x^{7}}{7}+\frac {\left (3 \left (3 A \,d^{2} e +B \,d^{3}\right ) a \,c^{2}+3 \left (A \,e^{3}+3 B d \,e^{2}\right ) a^{2} c \right ) x^{6}}{6}+\frac {\left (3 A \,d^{3} a \,c^{2}+3 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a^{2} c +B \,e^{3} a^{3}\right ) x^{5}}{5}+\frac {\left (3 \left (3 A \,d^{2} e +B \,d^{3}\right ) a^{2} c +\left (A \,e^{3}+3 B d \,e^{2}\right ) a^{3}\right ) x^{4}}{4}+\frac {\left (3 A \,d^{3} a^{2} c +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a^{3}\right ) x^{3}}{3}+\frac {\left (3 A \,d^{2} e +B \,d^{3}\right ) a^{3} x^{2}}{2}+A \,d^{3} a^{3} x\)	\(353\)
norman	\(\frac {B \,e^{3} c^{3} x^{11}}{11}+\left (\frac {1}{10} A \,c^{3} e^{3}+\frac {3}{10} B \,c^{3} d \,e^{2}\right ) x^{10}+\left (\frac {1}{3} A \,c^{3} d \,e^{2}+\frac {1}{3} B \,e^{3} a \,c^{2}+\frac {1}{3} B \,c^{3} d^{2} e \right ) x^{9}+\left (\frac {3}{8} A a \,c^{2} e^{3}+\frac {3}{8} A \,c^{3} d^{2} e +\frac {9}{8} B a \,c^{2} d \,e^{2}+\frac {1}{8} B \,c^{3} d^{3}\right ) x^{8}+\left (\frac {9}{7} A a \,c^{2} d \,e^{2}+\frac {1}{7} A \,c^{3} d^{3}+\frac {3}{7} B \,e^{3} a^{2} c +\frac {9}{7} B a \,c^{2} d^{2} e \right ) x^{7}+\left (\frac {1}{2} A \,a^{2} c \,e^{3}+\frac {3}{2} A a \,c^{2} d^{2} e +\frac {3}{2} B \,a^{2} c d \,e^{2}+\frac {1}{2} B a \,c^{2} d^{3}\right ) x^{6}+\left (\frac {9}{5} A \,a^{2} c d \,e^{2}+\frac {3}{5} A \,d^{3} a \,c^{2}+\frac {1}{5} B \,e^{3} a^{3}+\frac {9}{5} B \,a^{2} c \,d^{2} e \right ) x^{5}+\left (\frac {1}{4} A \,a^{3} e^{3}+\frac {9}{4} A \,a^{2} c \,d^{2} e +\frac {3}{4} B \,a^{3} d \,e^{2}+\frac {3}{4} B \,a^{2} c \,d^{3}\right ) x^{4}+\left (A \,a^{3} d \,e^{2}+A \,d^{3} a^{2} c +B \,a^{3} d^{2} e \right ) x^{3}+\left (\frac {3}{2} A \,a^{3} d^{2} e +\frac {1}{2} a^{3} B \,d^{3}\right ) x^{2}+A \,d^{3} a^{3} x\)	\(367\)
gosper	\(\frac {1}{3} x^{9} A \,c^{3} d \,e^{2}+\frac {1}{3} x^{9} B \,e^{3} a \,c^{2}+A \,a^{3} d \,e^{2} x^{3}+A \,a^{2} c \,d^{3} x^{3}+\frac {1}{4} x^{4} A \,a^{3} e^{3}+\frac {1}{2} x^{2} a^{3} B \,d^{3}+\frac {1}{2} x^{6} A \,a^{2} c \,e^{3}+\frac {1}{2} x^{6} B a \,c^{2} d^{3}+\frac {3}{8} x^{8} A \,c^{3} d^{2} e +\frac {3}{7} x^{7} B \,e^{3} a^{2} c +\frac {3}{5} x^{5} A \,d^{3} a \,c^{2}+\frac {3}{4} x^{4} B \,a^{3} d \,e^{2}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{3}+\frac {3}{2} x^{2} A \,a^{3} d^{2} e +\frac {3}{10} x^{10} B \,c^{3} d \,e^{2}+\frac {1}{3} x^{9} B \,c^{3} d^{2} e +\frac {3}{8} x^{8} A a \,c^{2} e^{3}+B \,a^{3} d^{2} e \,x^{3}+A \,d^{3} a^{3} x +\frac {1}{10} x^{10} A \,c^{3} e^{3}+\frac {1}{8} x^{8} B \,c^{3} d^{3}+\frac {1}{7} x^{7} A \,c^{3} d^{3}+\frac {9}{7} x^{7} B a \,c^{2} d^{2} e +\frac {9}{7} x^{7} A a \,c^{2} d \,e^{2}+\frac {9}{5} x^{5} B \,a^{2} c \,d^{2} e +\frac {3}{2} x^{6} A a \,c^{2} d^{2} e +\frac {9}{8} x^{8} B a \,c^{2} d \,e^{2}+\frac {9}{5} x^{5} A \,a^{2} c d \,e^{2}+\frac {3}{2} x^{6} B \,a^{2} c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a^{3}+\frac {9}{4} x^{4} A \,a^{2} c \,d^{2} e +\frac {1}{11} B \,e^{3} c^{3} x^{11}\)	\(412\)
risch	\(\frac {1}{3} x^{9} A \,c^{3} d \,e^{2}+\frac {1}{3} x^{9} B \,e^{3} a \,c^{2}+A \,a^{3} d \,e^{2} x^{3}+A \,a^{2} c \,d^{3} x^{3}+\frac {1}{4} x^{4} A \,a^{3} e^{3}+\frac {1}{2} x^{2} a^{3} B \,d^{3}+\frac {1}{2} x^{6} A \,a^{2} c \,e^{3}+\frac {1}{2} x^{6} B a \,c^{2} d^{3}+\frac {3}{8} x^{8} A \,c^{3} d^{2} e +\frac {3}{7} x^{7} B \,e^{3} a^{2} c +\frac {3}{5} x^{5} A \,d^{3} a \,c^{2}+\frac {3}{4} x^{4} B \,a^{3} d \,e^{2}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{3}+\frac {3}{2} x^{2} A \,a^{3} d^{2} e +\frac {3}{10} x^{10} B \,c^{3} d \,e^{2}+\frac {1}{3} x^{9} B \,c^{3} d^{2} e +\frac {3}{8} x^{8} A a \,c^{2} e^{3}+B \,a^{3} d^{2} e \,x^{3}+A \,d^{3} a^{3} x +\frac {1}{10} x^{10} A \,c^{3} e^{3}+\frac {1}{8} x^{8} B \,c^{3} d^{3}+\frac {1}{7} x^{7} A \,c^{3} d^{3}+\frac {9}{7} x^{7} B a \,c^{2} d^{2} e +\frac {9}{7} x^{7} A a \,c^{2} d \,e^{2}+\frac {9}{5} x^{5} B \,a^{2} c \,d^{2} e +\frac {3}{2} x^{6} A a \,c^{2} d^{2} e +\frac {9}{8} x^{8} B a \,c^{2} d \,e^{2}+\frac {9}{5} x^{5} A \,a^{2} c d \,e^{2}+\frac {3}{2} x^{6} B \,a^{2} c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a^{3}+\frac {9}{4} x^{4} A \,a^{2} c \,d^{2} e +\frac {1}{11} B \,e^{3} c^{3} x^{11}\)	\(412\)
parallelrisch	\(\frac {1}{3} x^{9} A \,c^{3} d \,e^{2}+\frac {1}{3} x^{9} B \,e^{3} a \,c^{2}+A \,a^{3} d \,e^{2} x^{3}+A \,a^{2} c \,d^{3} x^{3}+\frac {1}{4} x^{4} A \,a^{3} e^{3}+\frac {1}{2} x^{2} a^{3} B \,d^{3}+\frac {1}{2} x^{6} A \,a^{2} c \,e^{3}+\frac {1}{2} x^{6} B a \,c^{2} d^{3}+\frac {3}{8} x^{8} A \,c^{3} d^{2} e +\frac {3}{7} x^{7} B \,e^{3} a^{2} c +\frac {3}{5} x^{5} A \,d^{3} a \,c^{2}+\frac {3}{4} x^{4} B \,a^{3} d \,e^{2}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{3}+\frac {3}{2} x^{2} A \,a^{3} d^{2} e +\frac {3}{10} x^{10} B \,c^{3} d \,e^{2}+\frac {1}{3} x^{9} B \,c^{3} d^{2} e +\frac {3}{8} x^{8} A a \,c^{2} e^{3}+B \,a^{3} d^{2} e \,x^{3}+A \,d^{3} a^{3} x +\frac {1}{10} x^{10} A \,c^{3} e^{3}+\frac {1}{8} x^{8} B \,c^{3} d^{3}+\frac {1}{7} x^{7} A \,c^{3} d^{3}+\frac {9}{7} x^{7} B a \,c^{2} d^{2} e +\frac {9}{7} x^{7} A a \,c^{2} d \,e^{2}+\frac {9}{5} x^{5} B \,a^{2} c \,d^{2} e +\frac {3}{2} x^{6} A a \,c^{2} d^{2} e +\frac {9}{8} x^{8} B a \,c^{2} d \,e^{2}+\frac {9}{5} x^{5} A \,a^{2} c d \,e^{2}+\frac {3}{2} x^{6} B \,a^{2} c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a^{3}+\frac {9}{4} x^{4} A \,a^{2} c \,d^{2} e +\frac {1}{11} B \,e^{3} c^{3} x^{11}\)	\(412\)
orering	\(\frac {x \left (840 B \,e^{3} c^{3} x^{10}+924 A \,c^{3} e^{3} x^{9}+2772 B \,c^{3} d \,e^{2} x^{9}+3080 A \,c^{3} d \,e^{2} x^{8}+3080 B a \,c^{2} e^{3} x^{8}+3080 B \,c^{3} d^{2} e \,x^{8}+3465 A a \,c^{2} e^{3} x^{7}+3465 A \,c^{3} d^{2} e \,x^{7}+10395 B a \,c^{2} d \,e^{2} x^{7}+1155 B \,c^{3} d^{3} x^{7}+11880 A a \,c^{2} d \,e^{2} x^{6}+1320 A \,c^{3} d^{3} x^{6}+3960 B \,a^{2} c \,e^{3} x^{6}+11880 B a \,c^{2} d^{2} e \,x^{6}+4620 A \,a^{2} c \,e^{3} x^{5}+13860 A a \,c^{2} d^{2} e \,x^{5}+13860 B \,a^{2} c d \,e^{2} x^{5}+4620 B a \,c^{2} d^{3} x^{5}+16632 A \,a^{2} c d \,e^{2} x^{4}+5544 A a \,c^{2} d^{3} x^{4}+1848 B \,a^{3} e^{3} x^{4}+16632 B \,a^{2} c \,d^{2} e \,x^{4}+2310 A \,a^{3} e^{3} x^{3}+20790 A \,a^{2} c \,d^{2} e \,x^{3}+6930 B \,a^{3} d \,e^{2} x^{3}+6930 B \,a^{2} c \,d^{3} x^{3}+9240 A \,a^{3} d \,e^{2} x^{2}+9240 A \,a^{2} c \,d^{3} x^{2}+9240 B \,a^{3} d^{2} e \,x^{2}+13860 A \,a^{3} d^{2} e x +4620 B \,a^{3} d^{3} x +9240 A \,d^{3} a^{3}\right )}{9240}\)	\(414\)

Input:

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

Output:

1/11*B*e^3*c^3*x^11+1/10*(A*e^3+3*B*d*e^2)*c^3*x^10+1/9*((3*A*d*e^2+3*B*d^ 
2*e)*c^3+3*B*e^3*a*c^2)*x^9+1/8*((3*A*d^2*e+B*d^3)*c^3+3*(A*e^3+3*B*d*e^2) 
*a*c^2)*x^8+1/7*(A*c^3*d^3+3*(3*A*d*e^2+3*B*d^2*e)*a*c^2+3*B*e^3*a^2*c)*x^ 
7+1/6*(3*(3*A*d^2*e+B*d^3)*a*c^2+3*(A*e^3+3*B*d*e^2)*a^2*c)*x^6+1/5*(3*A*d 
^3*a*c^2+3*(3*A*d*e^2+3*B*d^2*e)*a^2*c+B*e^3*a^3)*x^5+1/4*(3*(3*A*d^2*e+B* 
d^3)*a^2*c+(A*e^3+3*B*d*e^2)*a^3)*x^4+1/3*(3*A*d^3*a^2*c+(3*A*d*e^2+3*B*d^ 
2*e)*a^3)*x^3+1/2*(3*A*d^2*e+B*d^3)*a^3*x^2+A*d^3*a^3*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.09 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^3 \, dx=\frac {1}{11} \, B c^{3} e^{3} x^{11} + \frac {1}{10} \, {\left (3 \, B c^{3} d e^{2} + A c^{3} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (B c^{3} d^{2} e + A c^{3} d e^{2} + B a c^{2} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{3} + 3 \, A c^{3} d^{2} e + 9 \, B a c^{2} d e^{2} + 3 \, A a c^{2} e^{3}\right )} x^{8} + A a^{3} d^{3} x + \frac {1}{7} \, {\left (A c^{3} d^{3} + 9 \, B a c^{2} d^{2} e + 9 \, A a c^{2} d e^{2} + 3 \, B a^{2} c e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B a c^{2} d^{3} + 3 \, A a c^{2} d^{2} e + 3 \, B a^{2} c d e^{2} + A a^{2} c e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A a c^{2} d^{3} + 9 \, B a^{2} c d^{2} e + 9 \, A a^{2} c d e^{2} + B a^{3} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{3} + 9 \, A a^{2} c d^{2} e + 3 \, B a^{3} d e^{2} + A a^{3} e^{3}\right )} x^{4} + {\left (A a^{2} c d^{3} + B a^{3} d^{2} e + A a^{3} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{3} + 3 \, A a^{3} d^{2} e\right )} x^{2} \] Input:

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

Output:

1/11*B*c^3*e^3*x^11 + 1/10*(3*B*c^3*d*e^2 + A*c^3*e^3)*x^10 + 1/3*(B*c^3*d 
^2*e + A*c^3*d*e^2 + B*a*c^2*e^3)*x^9 + 1/8*(B*c^3*d^3 + 3*A*c^3*d^2*e + 9 
*B*a*c^2*d*e^2 + 3*A*a*c^2*e^3)*x^8 + A*a^3*d^3*x + 1/7*(A*c^3*d^3 + 9*B*a 
*c^2*d^2*e + 9*A*a*c^2*d*e^2 + 3*B*a^2*c*e^3)*x^7 + 1/2*(B*a*c^2*d^3 + 3*A 
*a*c^2*d^2*e + 3*B*a^2*c*d*e^2 + A*a^2*c*e^3)*x^6 + 1/5*(3*A*a*c^2*d^3 + 9 
*B*a^2*c*d^2*e + 9*A*a^2*c*d*e^2 + B*a^3*e^3)*x^5 + 1/4*(3*B*a^2*c*d^3 + 9 
*A*a^2*c*d^2*e + 3*B*a^3*d*e^2 + A*a^3*e^3)*x^4 + (A*a^2*c*d^3 + B*a^3*d^2 
*e + A*a^3*d*e^2)*x^3 + 1/2*(B*a^3*d^3 + 3*A*a^3*d^2*e)*x^2
 

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.30 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^3 \, dx=A a^{3} d^{3} x + \frac {B c^{3} e^{3} x^{11}}{11} + x^{10} \left (\frac {A c^{3} e^{3}}{10} + \frac {3 B c^{3} d e^{2}}{10}\right ) + x^{9} \left (\frac {A c^{3} d e^{2}}{3} + \frac {B a c^{2} e^{3}}{3} + \frac {B c^{3} d^{2} e}{3}\right ) + x^{8} \cdot \left (\frac {3 A a c^{2} e^{3}}{8} + \frac {3 A c^{3} d^{2} e}{8} + \frac {9 B a c^{2} d e^{2}}{8} + \frac {B c^{3} d^{3}}{8}\right ) + x^{7} \cdot \left (\frac {9 A a c^{2} d e^{2}}{7} + \frac {A c^{3} d^{3}}{7} + \frac {3 B a^{2} c e^{3}}{7} + \frac {9 B a c^{2} d^{2} e}{7}\right ) + x^{6} \left (\frac {A a^{2} c e^{3}}{2} + \frac {3 A a c^{2} d^{2} e}{2} + \frac {3 B a^{2} c d e^{2}}{2} + \frac {B a c^{2} d^{3}}{2}\right ) + x^{5} \cdot \left (\frac {9 A a^{2} c d e^{2}}{5} + \frac {3 A a c^{2} d^{3}}{5} + \frac {B a^{3} e^{3}}{5} + \frac {9 B a^{2} c d^{2} e}{5}\right ) + x^{4} \left (\frac {A a^{3} e^{3}}{4} + \frac {9 A a^{2} c d^{2} e}{4} + \frac {3 B a^{3} d e^{2}}{4} + \frac {3 B a^{2} c d^{3}}{4}\right ) + x^{3} \left (A a^{3} d e^{2} + A a^{2} c d^{3} + B a^{3} d^{2} e\right ) + x^{2} \cdot \left (\frac {3 A a^{3} d^{2} e}{2} + \frac {B a^{3} d^{3}}{2}\right ) \] Input:

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

Output:

A*a**3*d**3*x + B*c**3*e**3*x**11/11 + x**10*(A*c**3*e**3/10 + 3*B*c**3*d* 
e**2/10) + x**9*(A*c**3*d*e**2/3 + B*a*c**2*e**3/3 + B*c**3*d**2*e/3) + x* 
*8*(3*A*a*c**2*e**3/8 + 3*A*c**3*d**2*e/8 + 9*B*a*c**2*d*e**2/8 + B*c**3*d 
**3/8) + x**7*(9*A*a*c**2*d*e**2/7 + A*c**3*d**3/7 + 3*B*a**2*c*e**3/7 + 9 
*B*a*c**2*d**2*e/7) + x**6*(A*a**2*c*e**3/2 + 3*A*a*c**2*d**2*e/2 + 3*B*a* 
*2*c*d*e**2/2 + B*a*c**2*d**3/2) + x**5*(9*A*a**2*c*d*e**2/5 + 3*A*a*c**2* 
d**3/5 + B*a**3*e**3/5 + 9*B*a**2*c*d**2*e/5) + x**4*(A*a**3*e**3/4 + 9*A* 
a**2*c*d**2*e/4 + 3*B*a**3*d*e**2/4 + 3*B*a**2*c*d**3/4) + x**3*(A*a**3*d* 
e**2 + A*a**2*c*d**3 + B*a**3*d**2*e) + x**2*(3*A*a**3*d**2*e/2 + B*a**3*d 
**3/2)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.09 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^3 \, dx=\frac {1}{11} \, B c^{3} e^{3} x^{11} + \frac {1}{10} \, {\left (3 \, B c^{3} d e^{2} + A c^{3} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (B c^{3} d^{2} e + A c^{3} d e^{2} + B a c^{2} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{3} + 3 \, A c^{3} d^{2} e + 9 \, B a c^{2} d e^{2} + 3 \, A a c^{2} e^{3}\right )} x^{8} + A a^{3} d^{3} x + \frac {1}{7} \, {\left (A c^{3} d^{3} + 9 \, B a c^{2} d^{2} e + 9 \, A a c^{2} d e^{2} + 3 \, B a^{2} c e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B a c^{2} d^{3} + 3 \, A a c^{2} d^{2} e + 3 \, B a^{2} c d e^{2} + A a^{2} c e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A a c^{2} d^{3} + 9 \, B a^{2} c d^{2} e + 9 \, A a^{2} c d e^{2} + B a^{3} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{3} + 9 \, A a^{2} c d^{2} e + 3 \, B a^{3} d e^{2} + A a^{3} e^{3}\right )} x^{4} + {\left (A a^{2} c d^{3} + B a^{3} d^{2} e + A a^{3} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{3} + 3 \, A a^{3} d^{2} e\right )} x^{2} \] Input:

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

Output:

1/11*B*c^3*e^3*x^11 + 1/10*(3*B*c^3*d*e^2 + A*c^3*e^3)*x^10 + 1/3*(B*c^3*d 
^2*e + A*c^3*d*e^2 + B*a*c^2*e^3)*x^9 + 1/8*(B*c^3*d^3 + 3*A*c^3*d^2*e + 9 
*B*a*c^2*d*e^2 + 3*A*a*c^2*e^3)*x^8 + A*a^3*d^3*x + 1/7*(A*c^3*d^3 + 9*B*a 
*c^2*d^2*e + 9*A*a*c^2*d*e^2 + 3*B*a^2*c*e^3)*x^7 + 1/2*(B*a*c^2*d^3 + 3*A 
*a*c^2*d^2*e + 3*B*a^2*c*d*e^2 + A*a^2*c*e^3)*x^6 + 1/5*(3*A*a*c^2*d^3 + 9 
*B*a^2*c*d^2*e + 9*A*a^2*c*d*e^2 + B*a^3*e^3)*x^5 + 1/4*(3*B*a^2*c*d^3 + 9 
*A*a^2*c*d^2*e + 3*B*a^3*d*e^2 + A*a^3*e^3)*x^4 + (A*a^2*c*d^3 + B*a^3*d^2 
*e + A*a^3*d*e^2)*x^3 + 1/2*(B*a^3*d^3 + 3*A*a^3*d^2*e)*x^2
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.23 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^3 \, dx=\frac {1}{11} \, B c^{3} e^{3} x^{11} + \frac {3}{10} \, B c^{3} d e^{2} x^{10} + \frac {1}{10} \, A c^{3} e^{3} x^{10} + \frac {1}{3} \, B c^{3} d^{2} e x^{9} + \frac {1}{3} \, A c^{3} d e^{2} x^{9} + \frac {1}{3} \, B a c^{2} e^{3} x^{9} + \frac {1}{8} \, B c^{3} d^{3} x^{8} + \frac {3}{8} \, A c^{3} d^{2} e x^{8} + \frac {9}{8} \, B a c^{2} d e^{2} x^{8} + \frac {3}{8} \, A a c^{2} e^{3} x^{8} + \frac {1}{7} \, A c^{3} d^{3} x^{7} + \frac {9}{7} \, B a c^{2} d^{2} e x^{7} + \frac {9}{7} \, A a c^{2} d e^{2} x^{7} + \frac {3}{7} \, B a^{2} c e^{3} x^{7} + \frac {1}{2} \, B a c^{2} d^{3} x^{6} + \frac {3}{2} \, A a c^{2} d^{2} e x^{6} + \frac {3}{2} \, B a^{2} c d e^{2} x^{6} + \frac {1}{2} \, A a^{2} c e^{3} x^{6} + \frac {3}{5} \, A a c^{2} d^{3} x^{5} + \frac {9}{5} \, B a^{2} c d^{2} e x^{5} + \frac {9}{5} \, A a^{2} c d e^{2} x^{5} + \frac {1}{5} \, B a^{3} e^{3} x^{5} + \frac {3}{4} \, B a^{2} c d^{3} x^{4} + \frac {9}{4} \, A a^{2} c d^{2} e x^{4} + \frac {3}{4} \, B a^{3} d e^{2} x^{4} + \frac {1}{4} \, A a^{3} e^{3} x^{4} + A a^{2} c d^{3} x^{3} + B a^{3} d^{2} e x^{3} + A a^{3} d e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{3} x^{2} + \frac {3}{2} \, A a^{3} d^{2} e x^{2} + A a^{3} d^{3} x \] Input:

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

Output:

1/11*B*c^3*e^3*x^11 + 3/10*B*c^3*d*e^2*x^10 + 1/10*A*c^3*e^3*x^10 + 1/3*B* 
c^3*d^2*e*x^9 + 1/3*A*c^3*d*e^2*x^9 + 1/3*B*a*c^2*e^3*x^9 + 1/8*B*c^3*d^3* 
x^8 + 3/8*A*c^3*d^2*e*x^8 + 9/8*B*a*c^2*d*e^2*x^8 + 3/8*A*a*c^2*e^3*x^8 + 
1/7*A*c^3*d^3*x^7 + 9/7*B*a*c^2*d^2*e*x^7 + 9/7*A*a*c^2*d*e^2*x^7 + 3/7*B* 
a^2*c*e^3*x^7 + 1/2*B*a*c^2*d^3*x^6 + 3/2*A*a*c^2*d^2*e*x^6 + 3/2*B*a^2*c* 
d*e^2*x^6 + 1/2*A*a^2*c*e^3*x^6 + 3/5*A*a*c^2*d^3*x^5 + 9/5*B*a^2*c*d^2*e* 
x^5 + 9/5*A*a^2*c*d*e^2*x^5 + 1/5*B*a^3*e^3*x^5 + 3/4*B*a^2*c*d^3*x^4 + 9/ 
4*A*a^2*c*d^2*e*x^4 + 3/4*B*a^3*d*e^2*x^4 + 1/4*A*a^3*e^3*x^4 + A*a^2*c*d^ 
3*x^3 + B*a^3*d^2*e*x^3 + A*a^3*d*e^2*x^3 + 1/2*B*a^3*d^3*x^2 + 3/2*A*a^3* 
d^2*e*x^2 + A*a^3*d^3*x
 

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.95 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^3 \, dx=x^5\,\left (\frac {B\,a^3\,e^3}{5}+\frac {9\,B\,a^2\,c\,d^2\,e}{5}+\frac {9\,A\,a^2\,c\,d\,e^2}{5}+\frac {3\,A\,a\,c^2\,d^3}{5}\right )+x^7\,\left (\frac {3\,B\,a^2\,c\,e^3}{7}+\frac {9\,B\,a\,c^2\,d^2\,e}{7}+\frac {9\,A\,a\,c^2\,d\,e^2}{7}+\frac {A\,c^3\,d^3}{7}\right )+\frac {a^2\,x^4\,\left (3\,B\,c\,d^3+9\,A\,c\,d^2\,e+3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{4}+\frac {c^2\,x^8\,\left (B\,c\,d^3+3\,A\,c\,d^2\,e+9\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{8}+a^2\,d\,x^3\,\left (A\,c\,d^2+B\,a\,d\,e+A\,a\,e^2\right )+\frac {c^2\,e\,x^9\,\left (B\,c\,d^2+A\,c\,d\,e+B\,a\,e^2\right )}{3}+\frac {a^3\,d^2\,x^2\,\left (3\,A\,e+B\,d\right )}{2}+\frac {c^3\,e^2\,x^{10}\,\left (A\,e+3\,B\,d\right )}{10}+\frac {a\,c\,x^6\,\left (B\,c\,d^3+3\,A\,c\,d^2\,e+3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{2}+A\,a^3\,d^3\,x+\frac {B\,c^3\,e^3\,x^{11}}{11} \] Input:

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

Output:

x^5*((B*a^3*e^3)/5 + (3*A*a*c^2*d^3)/5 + (9*A*a^2*c*d*e^2)/5 + (9*B*a^2*c* 
d^2*e)/5) + x^7*((A*c^3*d^3)/7 + (3*B*a^2*c*e^3)/7 + (9*A*a*c^2*d*e^2)/7 + 
 (9*B*a*c^2*d^2*e)/7) + (a^2*x^4*(A*a*e^3 + 3*B*c*d^3 + 3*B*a*d*e^2 + 9*A* 
c*d^2*e))/4 + (c^2*x^8*(3*A*a*e^3 + B*c*d^3 + 9*B*a*d*e^2 + 3*A*c*d^2*e))/ 
8 + a^2*d*x^3*(A*a*e^2 + A*c*d^2 + B*a*d*e) + (c^2*e*x^9*(B*a*e^2 + B*c*d^ 
2 + A*c*d*e))/3 + (a^3*d^2*x^2*(3*A*e + B*d))/2 + (c^3*e^2*x^10*(A*e + 3*B 
*d))/10 + (a*c*x^6*(A*a*e^3 + B*c*d^3 + 3*B*a*d*e^2 + 3*A*c*d^2*e))/2 + A* 
a^3*d^3*x + (B*c^3*e^3*x^11)/11
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.22 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^3 \, dx=\frac {x \left (840 b \,c^{3} e^{3} x^{10}+924 a \,c^{3} e^{3} x^{9}+2772 b \,c^{3} d \,e^{2} x^{9}+3080 a b \,c^{2} e^{3} x^{8}+3080 a \,c^{3} d \,e^{2} x^{8}+3080 b \,c^{3} d^{2} e \,x^{8}+3465 a^{2} c^{2} e^{3} x^{7}+10395 a b \,c^{2} d \,e^{2} x^{7}+3465 a \,c^{3} d^{2} e \,x^{7}+1155 b \,c^{3} d^{3} x^{7}+3960 a^{2} b c \,e^{3} x^{6}+11880 a^{2} c^{2} d \,e^{2} x^{6}+11880 a b \,c^{2} d^{2} e \,x^{6}+1320 a \,c^{3} d^{3} x^{6}+4620 a^{3} c \,e^{3} x^{5}+13860 a^{2} b c d \,e^{2} x^{5}+13860 a^{2} c^{2} d^{2} e \,x^{5}+4620 a b \,c^{2} d^{3} x^{5}+1848 a^{3} b \,e^{3} x^{4}+16632 a^{3} c d \,e^{2} x^{4}+16632 a^{2} b c \,d^{2} e \,x^{4}+5544 a^{2} c^{2} d^{3} x^{4}+2310 a^{4} e^{3} x^{3}+6930 a^{3} b d \,e^{2} x^{3}+20790 a^{3} c \,d^{2} e \,x^{3}+6930 a^{2} b c \,d^{3} x^{3}+9240 a^{4} d \,e^{2} x^{2}+9240 a^{3} b \,d^{2} e \,x^{2}+9240 a^{3} c \,d^{3} x^{2}+13860 a^{4} d^{2} e x +4620 a^{3} b \,d^{3} x +9240 a^{4} d^{3}\right )}{9240} \] Input:

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

Output:

(x*(9240*a**4*d**3 + 13860*a**4*d**2*e*x + 9240*a**4*d*e**2*x**2 + 2310*a* 
*4*e**3*x**3 + 4620*a**3*b*d**3*x + 9240*a**3*b*d**2*e*x**2 + 6930*a**3*b* 
d*e**2*x**3 + 1848*a**3*b*e**3*x**4 + 9240*a**3*c*d**3*x**2 + 20790*a**3*c 
*d**2*e*x**3 + 16632*a**3*c*d*e**2*x**4 + 4620*a**3*c*e**3*x**5 + 6930*a** 
2*b*c*d**3*x**3 + 16632*a**2*b*c*d**2*e*x**4 + 13860*a**2*b*c*d*e**2*x**5 
+ 3960*a**2*b*c*e**3*x**6 + 5544*a**2*c**2*d**3*x**4 + 13860*a**2*c**2*d** 
2*e*x**5 + 11880*a**2*c**2*d*e**2*x**6 + 3465*a**2*c**2*e**3*x**7 + 4620*a 
*b*c**2*d**3*x**5 + 11880*a*b*c**2*d**2*e*x**6 + 10395*a*b*c**2*d*e**2*x** 
7 + 3080*a*b*c**2*e**3*x**8 + 1320*a*c**3*d**3*x**6 + 3465*a*c**3*d**2*e*x 
**7 + 3080*a*c**3*d*e**2*x**8 + 924*a*c**3*e**3*x**9 + 1155*b*c**3*d**3*x* 
*7 + 3080*b*c**3*d**2*e*x**8 + 2772*b*c**3*d*e**2*x**9 + 840*b*c**3*e**3*x 
**10))/9240