\(\int (A+B x) (d+e x)^5 (a+c x^2)^3 \, dx\) [62]

Optimal result

Integrand size = 22, antiderivative size = 334 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right )^3 \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right )^3 (d+e x)^6}{6 e^8}+\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^7}{7 e^8}-\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^8}{8 e^8}-\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^9}{9 e^8}-\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{10}}{10 e^8}+\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{11}}{11 e^8}-\frac {c^3 (7 B d-A e) (d+e x)^{12}}{12 e^8}+\frac {B c^3 (d+e x)^{13}}{13 e^8} \] Output:

-1/6*(-A*e+B*d)*(a*e^2+c*d^2)^3*(e*x+d)^6/e^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*x+d)^7/e^8-3/8*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*x+d)^8/e^8-1/9*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)^9/e^8-1/10*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)^10/e^8+3/11*c^2*(-2*A*c 
*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^11/e^8-1/12*c^3*(-A*e+7*B*d)*(e*x+d)^12/e^ 
8+1/13*B*c^3*(e*x+d)^13/e^8
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.62 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right )^3 \, dx=a^3 A d^5 x+\frac {1}{2} a^3 d^4 (B d+5 A e) x^2+\frac {1}{3} a^2 d^3 \left (3 A c d^2+5 a B d e+10 a A e^2\right ) x^3+\frac {1}{4} a^2 d^2 \left (3 B c d^3+15 A c d^2 e+10 a B d e^2+10 a A e^3\right ) x^4+\frac {1}{5} a d \left (5 a B d e \left (3 c d^2+2 a e^2\right )+A \left (3 c^2 d^4+30 a c d^2 e^2+5 a^2 e^4\right )\right ) x^5+\frac {1}{6} a \left (A e \left (15 c^2 d^4+30 a c d^2 e^2+a^2 e^4\right )+B \left (3 c^2 d^5+30 a c d^3 e^2+5 a^2 d e^4\right )\right ) x^6+\frac {1}{7} \left (a B e \left (15 c^2 d^4+30 a c d^2 e^2+a^2 e^4\right )+A c d \left (c^2 d^4+30 a c d^2 e^2+15 a^2 e^4\right )\right ) x^7+\frac {1}{8} c \left (A e \left (5 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )+B \left (c^2 d^5+30 a c d^3 e^2+15 a^2 d e^4\right )\right ) x^8+\frac {1}{9} c e \left (5 A c d e \left (2 c d^2+3 a e^2\right )+B \left (5 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) x^9+\frac {1}{10} c^2 e^2 \left (10 B c d^3+10 A c d^2 e+15 a B d e^2+3 a A e^3\right ) x^{10}+\frac {1}{11} c^2 e^3 \left (10 B c d^2+5 A c d e+3 a B e^2\right ) x^{11}+\frac {1}{12} c^3 e^4 (5 B d+A e) x^{12}+\frac {1}{13} B c^3 e^5 x^{13} \] Input:

Integrate[(A + B*x)*(d + e*x)^5*(a + c*x^2)^3,x]
 

Output:

a^3*A*d^5*x + (a^3*d^4*(B*d + 5*A*e)*x^2)/2 + (a^2*d^3*(3*A*c*d^2 + 5*a*B* 
d*e + 10*a*A*e^2)*x^3)/3 + (a^2*d^2*(3*B*c*d^3 + 15*A*c*d^2*e + 10*a*B*d*e 
^2 + 10*a*A*e^3)*x^4)/4 + (a*d*(5*a*B*d*e*(3*c*d^2 + 2*a*e^2) + A*(3*c^2*d 
^4 + 30*a*c*d^2*e^2 + 5*a^2*e^4))*x^5)/5 + (a*(A*e*(15*c^2*d^4 + 30*a*c*d^ 
2*e^2 + a^2*e^4) + B*(3*c^2*d^5 + 30*a*c*d^3*e^2 + 5*a^2*d*e^4))*x^6)/6 + 
((a*B*e*(15*c^2*d^4 + 30*a*c*d^2*e^2 + a^2*e^4) + A*c*d*(c^2*d^4 + 30*a*c* 
d^2*e^2 + 15*a^2*e^4))*x^7)/7 + (c*(A*e*(5*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^ 
2*e^4) + B*(c^2*d^5 + 30*a*c*d^3*e^2 + 15*a^2*d*e^4))*x^8)/8 + (c*e*(5*A*c 
*d*e*(2*c*d^2 + 3*a*e^2) + B*(5*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*x^9 
)/9 + (c^2*e^2*(10*B*c*d^3 + 10*A*c*d^2*e + 15*a*B*d*e^2 + 3*a*A*e^3)*x^10 
)/10 + (c^2*e^3*(10*B*c*d^2 + 5*A*c*d*e + 3*a*B*e^2)*x^11)/11 + (c^3*e^4*( 
5*B*d + A*e)*x^12)/12 + (B*c^3*e^5*x^13)/13
 

Rubi [A] (verified)

Time = 0.85 (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.

\(\displaystyle \int \left (a+c x^2\right )^3 (A+B x) (d+e x)^5 \, dx\)

\(\Big \downarrow \) 652

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c (d+e x)^9 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{9 e^8}+\frac {3 c^2 (d+e x)^{11} \left (a B e^2-2 A c d e+7 B c d^2\right )}{11 e^8}-\frac {c^2 (d+e x)^{10} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{10 e^8}+\frac {(d+e x)^7 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8}-\frac {(d+e x)^6 \left (a e^2+c d^2\right )^3 (B d-A e)}{6 e^8}-\frac {3 c (d+e x)^8 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{8 e^8}-\frac {c^3 (d+e x)^{12} (7 B d-A e)}{12 e^8}+\frac {B c^3 (d+e x)^{13}}{13 e^8}\)

Input:

Int[(A + B*x)*(d + e*x)^5*(a + c*x^2)^3,x]
 

Output:

-1/6*((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^6)/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)^7)/(7*e^8) - (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)*(d + e*x)^8)/(8*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)^9)/(9*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)^10)/(10*e^8) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*e 
+ a*B*e^2)*(d + e*x)^11)/(11*e^8) - (c^3*(7*B*d - A*e)*(d + e*x)^12)/(12*e 
^8) + (B*c^3*(d + e*x)^13)/(13*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 = 557, normalized size of antiderivative = 1.67

method	result	size
default	\(\frac {B \,e^{5} c^{3} x^{13}}{13}+\frac {\left (A \,e^{5}+5 B d \,e^{4}\right ) c^{3} x^{12}}{12}+\frac {\left (\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) c^{3}+3 B \,e^{5} a \,c^{2}\right ) x^{11}}{11}+\frac {\left (\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) c^{3}+3 \left (A \,e^{5}+5 B d \,e^{4}\right ) a \,c^{2}\right ) x^{10}}{10}+\frac {\left (\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) c^{3}+3 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a \,c^{2}+3 B \,e^{5} a^{2} c \right ) x^{9}}{9}+\frac {\left (\left (5 A \,d^{4} e +B \,d^{5}\right ) c^{3}+3 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a \,c^{2}+3 \left (A \,e^{5}+5 B d \,e^{4}\right ) a^{2} c \right ) x^{8}}{8}+\frac {\left (A \,d^{5} c^{3}+3 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a \,c^{2}+3 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{2} c +B \,e^{5} a^{3}\right ) x^{7}}{7}+\frac {\left (3 \left (5 A \,d^{4} e +B \,d^{5}\right ) a \,c^{2}+3 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{2} c +\left (A \,e^{5}+5 B d \,e^{4}\right ) a^{3}\right ) x^{6}}{6}+\frac {\left (3 A \,d^{5} a \,c^{2}+3 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a^{2} c +\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{3}\right ) x^{5}}{5}+\frac {\left (3 \left (5 A \,d^{4} e +B \,d^{5}\right ) a^{2} c +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{3}\right ) x^{4}}{4}+\frac {\left (3 A \,d^{5} a^{2} c +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a^{3}\right ) x^{3}}{3}+\frac {\left (5 A \,d^{4} e +B \,d^{5}\right ) a^{3} x^{2}}{2}+A \,d^{5} a^{3} x\)	\(557\)
norman	\(\frac {B \,e^{5} c^{3} x^{13}}{13}+\left (\frac {1}{12} A \,c^{3} e^{5}+\frac {5}{12} B \,c^{3} d \,e^{4}\right ) x^{12}+\left (\frac {5}{11} A \,c^{3} d \,e^{4}+\frac {3}{11} B \,e^{5} a \,c^{2}+\frac {10}{11} B \,c^{3} d^{2} e^{3}\right ) x^{11}+\left (\frac {3}{10} A a \,c^{2} e^{5}+A \,c^{3} d^{2} e^{3}+\frac {3}{2} B a \,c^{2} d \,e^{4}+B \,c^{3} d^{3} e^{2}\right ) x^{10}+\left (\frac {5}{3} A a \,c^{2} d \,e^{4}+\frac {10}{9} A \,c^{3} d^{3} e^{2}+\frac {1}{3} B \,e^{5} a^{2} c +\frac {10}{3} B a \,c^{2} d^{2} e^{3}+\frac {5}{9} B \,c^{3} d^{4} e \right ) x^{9}+\left (\frac {3}{8} A \,a^{2} c \,e^{5}+\frac {15}{4} A a \,c^{2} d^{2} e^{3}+\frac {5}{8} A \,c^{3} d^{4} e +\frac {15}{8} B \,a^{2} c d \,e^{4}+\frac {15}{4} B a \,c^{2} d^{3} e^{2}+\frac {1}{8} B \,c^{3} d^{5}\right ) x^{8}+\left (\frac {15}{7} A \,a^{2} c d \,e^{4}+\frac {30}{7} A a \,c^{2} d^{3} e^{2}+\frac {1}{7} A \,d^{5} c^{3}+\frac {1}{7} B \,e^{5} a^{3}+\frac {30}{7} B \,a^{2} c \,d^{2} e^{3}+\frac {15}{7} B a \,c^{2} d^{4} e \right ) x^{7}+\left (\frac {1}{6} A \,a^{3} e^{5}+5 A \,a^{2} c \,d^{2} e^{3}+\frac {5}{2} A a \,c^{2} d^{4} e +\frac {5}{6} B \,a^{3} d \,e^{4}+5 B \,a^{2} c \,d^{3} e^{2}+\frac {1}{2} B a \,c^{2} d^{5}\right ) x^{6}+\left (A \,a^{3} d \,e^{4}+6 A \,a^{2} c \,d^{3} e^{2}+\frac {3}{5} A \,d^{5} a \,c^{2}+2 B \,a^{3} d^{2} e^{3}+3 B \,a^{2} c \,d^{4} e \right ) x^{5}+\left (\frac {5}{2} A \,a^{3} d^{2} e^{3}+\frac {15}{4} A \,a^{2} c \,d^{4} e +\frac {5}{2} B \,a^{3} d^{3} e^{2}+\frac {3}{4} B \,a^{2} c \,d^{5}\right ) x^{4}+\left (\frac {10}{3} A \,a^{3} d^{3} e^{2}+A \,d^{5} a^{2} c +\frac {5}{3} B \,a^{3} d^{4} e \right ) x^{3}+\left (\frac {5}{2} A \,a^{3} d^{4} e +\frac {1}{2} B \,a^{3} d^{5}\right ) x^{2}+A \,d^{5} a^{3} x\)	\(576\)
gosper	\(\frac {5}{3} x^{9} A a \,c^{2} d \,e^{4}+x^{5} A \,a^{3} d \,e^{4}+\frac {3}{5} x^{5} A \,d^{5} a \,c^{2}+2 x^{5} B \,a^{3} d^{2} e^{3}+\frac {5}{2} x^{4} A \,a^{3} d^{2} e^{3}+\frac {15}{7} x^{7} B a \,c^{2} d^{4} e +\frac {5}{2} x^{6} A a \,c^{2} d^{4} e +\frac {5}{12} x^{12} B \,c^{3} d \,e^{4}+\frac {5}{11} x^{11} A \,c^{3} d \,e^{4}+\frac {15}{4} x^{8} A a \,c^{2} d^{2} e^{3}+\frac {3}{8} x^{8} A \,a^{2} c \,e^{5}+\frac {5}{8} x^{8} A \,c^{3} d^{4} e +\frac {5}{6} x^{6} B \,a^{3} d \,e^{4}+\frac {1}{2} x^{6} B a \,c^{2} d^{5}+5 x^{6} B \,a^{2} c \,d^{3} e^{2}+6 x^{5} A \,a^{2} c \,d^{3} e^{2}+\frac {5}{9} x^{9} B \,c^{3} d^{4} e +\frac {3}{10} x^{10} A a \,c^{2} e^{5}+\frac {10}{3} x^{9} B a \,c^{2} d^{2} e^{3}+\frac {1}{13} B \,e^{5} c^{3} x^{13}+\frac {15}{4} x^{4} A \,a^{2} c \,d^{4} e +\frac {15}{7} x^{7} A \,a^{2} c d \,e^{4}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{5}+\frac {10}{3} x^{3} A \,a^{3} d^{3} e^{2}+x^{3} A \,d^{5} a^{2} c +\frac {5}{3} x^{3} B \,a^{3} d^{4} e +\frac {5}{2} x^{2} A \,a^{3} d^{4} e +\frac {1}{2} x^{2} B \,a^{3} d^{5}+\frac {3}{11} x^{11} B \,e^{5} a \,c^{2}+\frac {10}{11} x^{11} B \,c^{3} d^{2} e^{3}+x^{10} A \,c^{3} d^{2} e^{3}+x^{10} B \,c^{3} d^{3} e^{2}+\frac {10}{9} x^{9} A \,c^{3} d^{3} e^{2}+\frac {1}{3} x^{9} B \,e^{5} a^{2} c +\frac {5}{2} x^{4} B \,a^{3} d^{3} e^{2}+3 x^{5} B \,a^{2} c \,d^{4} e +A \,d^{5} a^{3} x +\frac {1}{12} x^{12} A \,c^{3} e^{5}+\frac {3}{2} x^{10} B a \,c^{2} d \,e^{4}+\frac {30}{7} x^{7} B \,a^{2} c \,d^{2} e^{3}+\frac {30}{7} x^{7} A a \,c^{2} d^{3} e^{2}+\frac {15}{8} x^{8} B \,a^{2} c d \,e^{4}+\frac {15}{4} x^{8} B a \,c^{2} d^{3} e^{2}+5 x^{6} A \,a^{2} c \,d^{2} e^{3}+\frac {1}{8} x^{8} B \,c^{3} d^{5}+\frac {1}{7} x^{7} A \,d^{5} c^{3}+\frac {1}{7} x^{7} B \,e^{5} a^{3}+\frac {1}{6} x^{6} A \,a^{3} e^{5}\)	\(659\)
risch	\(\frac {5}{3} x^{9} A a \,c^{2} d \,e^{4}+x^{5} A \,a^{3} d \,e^{4}+\frac {3}{5} x^{5} A \,d^{5} a \,c^{2}+2 x^{5} B \,a^{3} d^{2} e^{3}+\frac {5}{2} x^{4} A \,a^{3} d^{2} e^{3}+\frac {15}{7} x^{7} B a \,c^{2} d^{4} e +\frac {5}{2} x^{6} A a \,c^{2} d^{4} e +\frac {5}{12} x^{12} B \,c^{3} d \,e^{4}+\frac {5}{11} x^{11} A \,c^{3} d \,e^{4}+\frac {15}{4} x^{8} A a \,c^{2} d^{2} e^{3}+\frac {3}{8} x^{8} A \,a^{2} c \,e^{5}+\frac {5}{8} x^{8} A \,c^{3} d^{4} e +\frac {5}{6} x^{6} B \,a^{3} d \,e^{4}+\frac {1}{2} x^{6} B a \,c^{2} d^{5}+5 x^{6} B \,a^{2} c \,d^{3} e^{2}+6 x^{5} A \,a^{2} c \,d^{3} e^{2}+\frac {5}{9} x^{9} B \,c^{3} d^{4} e +\frac {3}{10} x^{10} A a \,c^{2} e^{5}+\frac {10}{3} x^{9} B a \,c^{2} d^{2} e^{3}+\frac {1}{13} B \,e^{5} c^{3} x^{13}+\frac {15}{4} x^{4} A \,a^{2} c \,d^{4} e +\frac {15}{7} x^{7} A \,a^{2} c d \,e^{4}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{5}+\frac {10}{3} x^{3} A \,a^{3} d^{3} e^{2}+x^{3} A \,d^{5} a^{2} c +\frac {5}{3} x^{3} B \,a^{3} d^{4} e +\frac {5}{2} x^{2} A \,a^{3} d^{4} e +\frac {1}{2} x^{2} B \,a^{3} d^{5}+\frac {3}{11} x^{11} B \,e^{5} a \,c^{2}+\frac {10}{11} x^{11} B \,c^{3} d^{2} e^{3}+x^{10} A \,c^{3} d^{2} e^{3}+x^{10} B \,c^{3} d^{3} e^{2}+\frac {10}{9} x^{9} A \,c^{3} d^{3} e^{2}+\frac {1}{3} x^{9} B \,e^{5} a^{2} c +\frac {5}{2} x^{4} B \,a^{3} d^{3} e^{2}+3 x^{5} B \,a^{2} c \,d^{4} e +A \,d^{5} a^{3} x +\frac {1}{12} x^{12} A \,c^{3} e^{5}+\frac {3}{2} x^{10} B a \,c^{2} d \,e^{4}+\frac {30}{7} x^{7} B \,a^{2} c \,d^{2} e^{3}+\frac {30}{7} x^{7} A a \,c^{2} d^{3} e^{2}+\frac {15}{8} x^{8} B \,a^{2} c d \,e^{4}+\frac {15}{4} x^{8} B a \,c^{2} d^{3} e^{2}+5 x^{6} A \,a^{2} c \,d^{2} e^{3}+\frac {1}{8} x^{8} B \,c^{3} d^{5}+\frac {1}{7} x^{7} A \,d^{5} c^{3}+\frac {1}{7} x^{7} B \,e^{5} a^{3}+\frac {1}{6} x^{6} A \,a^{3} e^{5}\)	\(659\)
parallelrisch	\(\frac {5}{3} x^{9} A a \,c^{2} d \,e^{4}+x^{5} A \,a^{3} d \,e^{4}+\frac {3}{5} x^{5} A \,d^{5} a \,c^{2}+2 x^{5} B \,a^{3} d^{2} e^{3}+\frac {5}{2} x^{4} A \,a^{3} d^{2} e^{3}+\frac {15}{7} x^{7} B a \,c^{2} d^{4} e +\frac {5}{2} x^{6} A a \,c^{2} d^{4} e +\frac {5}{12} x^{12} B \,c^{3} d \,e^{4}+\frac {5}{11} x^{11} A \,c^{3} d \,e^{4}+\frac {15}{4} x^{8} A a \,c^{2} d^{2} e^{3}+\frac {3}{8} x^{8} A \,a^{2} c \,e^{5}+\frac {5}{8} x^{8} A \,c^{3} d^{4} e +\frac {5}{6} x^{6} B \,a^{3} d \,e^{4}+\frac {1}{2} x^{6} B a \,c^{2} d^{5}+5 x^{6} B \,a^{2} c \,d^{3} e^{2}+6 x^{5} A \,a^{2} c \,d^{3} e^{2}+\frac {5}{9} x^{9} B \,c^{3} d^{4} e +\frac {3}{10} x^{10} A a \,c^{2} e^{5}+\frac {10}{3} x^{9} B a \,c^{2} d^{2} e^{3}+\frac {1}{13} B \,e^{5} c^{3} x^{13}+\frac {15}{4} x^{4} A \,a^{2} c \,d^{4} e +\frac {15}{7} x^{7} A \,a^{2} c d \,e^{4}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{5}+\frac {10}{3} x^{3} A \,a^{3} d^{3} e^{2}+x^{3} A \,d^{5} a^{2} c +\frac {5}{3} x^{3} B \,a^{3} d^{4} e +\frac {5}{2} x^{2} A \,a^{3} d^{4} e +\frac {1}{2} x^{2} B \,a^{3} d^{5}+\frac {3}{11} x^{11} B \,e^{5} a \,c^{2}+\frac {10}{11} x^{11} B \,c^{3} d^{2} e^{3}+x^{10} A \,c^{3} d^{2} e^{3}+x^{10} B \,c^{3} d^{3} e^{2}+\frac {10}{9} x^{9} A \,c^{3} d^{3} e^{2}+\frac {1}{3} x^{9} B \,e^{5} a^{2} c +\frac {5}{2} x^{4} B \,a^{3} d^{3} e^{2}+3 x^{5} B \,a^{2} c \,d^{4} e +A \,d^{5} a^{3} x +\frac {1}{12} x^{12} A \,c^{3} e^{5}+\frac {3}{2} x^{10} B a \,c^{2} d \,e^{4}+\frac {30}{7} x^{7} B \,a^{2} c \,d^{2} e^{3}+\frac {30}{7} x^{7} A a \,c^{2} d^{3} e^{2}+\frac {15}{8} x^{8} B \,a^{2} c d \,e^{4}+\frac {15}{4} x^{8} B a \,c^{2} d^{3} e^{2}+5 x^{6} A \,a^{2} c \,d^{2} e^{3}+\frac {1}{8} x^{8} B \,c^{3} d^{5}+\frac {1}{7} x^{7} A \,d^{5} c^{3}+\frac {1}{7} x^{7} B \,e^{5} a^{3}+\frac {1}{6} x^{6} A \,a^{3} e^{5}\)	\(659\)
orering	\(\frac {x \left (27720 B \,e^{5} c^{3} x^{12}+30030 A \,c^{3} e^{5} x^{11}+150150 B \,c^{3} d \,e^{4} x^{11}+163800 A \,c^{3} d \,e^{4} x^{10}+98280 B a \,c^{2} e^{5} x^{10}+327600 B \,c^{3} d^{2} e^{3} x^{10}+108108 A a \,c^{2} e^{5} x^{9}+360360 A \,c^{3} d^{2} e^{3} x^{9}+540540 B a \,c^{2} d \,e^{4} x^{9}+360360 B \,c^{3} d^{3} e^{2} x^{9}+600600 A a \,c^{2} d \,e^{4} x^{8}+400400 A \,c^{3} d^{3} e^{2} x^{8}+120120 B \,a^{2} c \,e^{5} x^{8}+1201200 B a \,c^{2} d^{2} e^{3} x^{8}+200200 B \,c^{3} d^{4} e \,x^{8}+135135 A \,a^{2} c \,e^{5} x^{7}+1351350 A a \,c^{2} d^{2} e^{3} x^{7}+225225 A \,c^{3} d^{4} e \,x^{7}+675675 B \,a^{2} c d \,e^{4} x^{7}+1351350 B a \,c^{2} d^{3} e^{2} x^{7}+45045 B \,c^{3} d^{5} x^{7}+772200 A \,a^{2} c d \,e^{4} x^{6}+1544400 A a \,c^{2} d^{3} e^{2} x^{6}+51480 A \,c^{3} d^{5} x^{6}+51480 B \,a^{3} e^{5} x^{6}+1544400 B \,a^{2} c \,d^{2} e^{3} x^{6}+772200 B a \,c^{2} d^{4} e \,x^{6}+60060 A \,a^{3} e^{5} x^{5}+1801800 A \,a^{2} c \,d^{2} e^{3} x^{5}+900900 A a \,c^{2} d^{4} e \,x^{5}+300300 B \,a^{3} d \,e^{4} x^{5}+1801800 B \,a^{2} c \,d^{3} e^{2} x^{5}+180180 B a \,c^{2} d^{5} x^{5}+360360 A \,a^{3} d \,e^{4} x^{4}+2162160 A \,a^{2} c \,d^{3} e^{2} x^{4}+216216 A a \,c^{2} d^{5} x^{4}+720720 B \,a^{3} d^{2} e^{3} x^{4}+1081080 B \,a^{2} c \,d^{4} e \,x^{4}+900900 A \,a^{3} d^{2} e^{3} x^{3}+1351350 A \,a^{2} c \,d^{4} e \,x^{3}+900900 B \,a^{3} d^{3} e^{2} x^{3}+270270 B \,a^{2} c \,d^{5} x^{3}+1201200 A \,a^{3} d^{3} e^{2} x^{2}+360360 A \,a^{2} c \,d^{5} x^{2}+600600 B \,a^{3} d^{4} e \,x^{2}+900900 A \,a^{3} d^{4} e x +180180 B \,a^{3} d^{5} x +360360 A \,d^{5} a^{3}\right )}{360360}\)	\(662\)

Input:

int((B*x+A)*(e*x+d)^5*(c*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/13*B*e^5*c^3*x^13+1/12*(A*e^5+5*B*d*e^4)*c^3*x^12+1/11*((5*A*d*e^4+10*B* 
d^2*e^3)*c^3+3*B*e^5*a*c^2)*x^11+1/10*((10*A*d^2*e^3+10*B*d^3*e^2)*c^3+3*( 
A*e^5+5*B*d*e^4)*a*c^2)*x^10+1/9*((10*A*d^3*e^2+5*B*d^4*e)*c^3+3*(5*A*d*e^ 
4+10*B*d^2*e^3)*a*c^2+3*B*e^5*a^2*c)*x^9+1/8*((5*A*d^4*e+B*d^5)*c^3+3*(10* 
A*d^2*e^3+10*B*d^3*e^2)*a*c^2+3*(A*e^5+5*B*d*e^4)*a^2*c)*x^8+1/7*(A*d^5*c^ 
3+3*(10*A*d^3*e^2+5*B*d^4*e)*a*c^2+3*(5*A*d*e^4+10*B*d^2*e^3)*a^2*c+B*e^5* 
a^3)*x^7+1/6*(3*(5*A*d^4*e+B*d^5)*a*c^2+3*(10*A*d^2*e^3+10*B*d^3*e^2)*a^2* 
c+(A*e^5+5*B*d*e^4)*a^3)*x^6+1/5*(3*A*d^5*a*c^2+3*(10*A*d^3*e^2+5*B*d^4*e) 
*a^2*c+(5*A*d*e^4+10*B*d^2*e^3)*a^3)*x^5+1/4*(3*(5*A*d^4*e+B*d^5)*a^2*c+(1 
0*A*d^2*e^3+10*B*d^3*e^2)*a^3)*x^4+1/3*(3*A*d^5*a^2*c+(10*A*d^3*e^2+5*B*d^ 
4*e)*a^3)*x^3+1/2*(5*A*d^4*e+B*d^5)*a^3*x^2+A*d^5*a^3*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.75 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right )^3 \, dx=\frac {1}{13} \, B c^{3} e^{5} x^{13} + \frac {1}{12} \, {\left (5 \, B c^{3} d e^{4} + A c^{3} e^{5}\right )} x^{12} + \frac {1}{11} \, {\left (10 \, B c^{3} d^{2} e^{3} + 5 \, A c^{3} d e^{4} + 3 \, B a c^{2} e^{5}\right )} x^{11} + \frac {1}{10} \, {\left (10 \, B c^{3} d^{3} e^{2} + 10 \, A c^{3} d^{2} e^{3} + 15 \, B a c^{2} d e^{4} + 3 \, A a c^{2} e^{5}\right )} x^{10} + A a^{3} d^{5} x + \frac {1}{9} \, {\left (5 \, B c^{3} d^{4} e + 10 \, A c^{3} d^{3} e^{2} + 30 \, B a c^{2} d^{2} e^{3} + 15 \, A a c^{2} d e^{4} + 3 \, B a^{2} c e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{5} + 5 \, A c^{3} d^{4} e + 30 \, B a c^{2} d^{3} e^{2} + 30 \, A a c^{2} d^{2} e^{3} + 15 \, B a^{2} c d e^{4} + 3 \, A a^{2} c e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d^{5} + 15 \, B a c^{2} d^{4} e + 30 \, A a c^{2} d^{3} e^{2} + 30 \, B a^{2} c d^{2} e^{3} + 15 \, A a^{2} c d e^{4} + B a^{3} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, B a c^{2} d^{5} + 15 \, A a c^{2} d^{4} e + 30 \, B a^{2} c d^{3} e^{2} + 30 \, A a^{2} c d^{2} e^{3} + 5 \, B a^{3} d e^{4} + A a^{3} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A a c^{2} d^{5} + 15 \, B a^{2} c d^{4} e + 30 \, A a^{2} c d^{3} e^{2} + 10 \, B a^{3} d^{2} e^{3} + 5 \, A a^{3} d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{5} + 15 \, A a^{2} c d^{4} e + 10 \, B a^{3} d^{3} e^{2} + 10 \, A a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d^{5} + 5 \, B a^{3} d^{4} e + 10 \, A a^{3} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{5} + 5 \, A a^{3} d^{4} e\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a)^3,x, algorithm="fricas")
 

Output:

1/13*B*c^3*e^5*x^13 + 1/12*(5*B*c^3*d*e^4 + A*c^3*e^5)*x^12 + 1/11*(10*B*c 
^3*d^2*e^3 + 5*A*c^3*d*e^4 + 3*B*a*c^2*e^5)*x^11 + 1/10*(10*B*c^3*d^3*e^2 
+ 10*A*c^3*d^2*e^3 + 15*B*a*c^2*d*e^4 + 3*A*a*c^2*e^5)*x^10 + A*a^3*d^5*x 
+ 1/9*(5*B*c^3*d^4*e + 10*A*c^3*d^3*e^2 + 30*B*a*c^2*d^2*e^3 + 15*A*a*c^2* 
d*e^4 + 3*B*a^2*c*e^5)*x^9 + 1/8*(B*c^3*d^5 + 5*A*c^3*d^4*e + 30*B*a*c^2*d 
^3*e^2 + 30*A*a*c^2*d^2*e^3 + 15*B*a^2*c*d*e^4 + 3*A*a^2*c*e^5)*x^8 + 1/7* 
(A*c^3*d^5 + 15*B*a*c^2*d^4*e + 30*A*a*c^2*d^3*e^2 + 30*B*a^2*c*d^2*e^3 + 
15*A*a^2*c*d*e^4 + B*a^3*e^5)*x^7 + 1/6*(3*B*a*c^2*d^5 + 15*A*a*c^2*d^4*e 
+ 30*B*a^2*c*d^3*e^2 + 30*A*a^2*c*d^2*e^3 + 5*B*a^3*d*e^4 + A*a^3*e^5)*x^6 
 + 1/5*(3*A*a*c^2*d^5 + 15*B*a^2*c*d^4*e + 30*A*a^2*c*d^3*e^2 + 10*B*a^3*d 
^2*e^3 + 5*A*a^3*d*e^4)*x^5 + 1/4*(3*B*a^2*c*d^5 + 15*A*a^2*c*d^4*e + 10*B 
*a^3*d^3*e^2 + 10*A*a^3*d^2*e^3)*x^4 + 1/3*(3*A*a^2*c*d^5 + 5*B*a^3*d^4*e 
+ 10*A*a^3*d^3*e^2)*x^3 + 1/2*(B*a^3*d^5 + 5*A*a^3*d^4*e)*x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (340) = 680\).

Time = 0.06 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.08 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right )^3 \, dx=A a^{3} d^{5} x + \frac {B c^{3} e^{5} x^{13}}{13} + x^{12} \left (\frac {A c^{3} e^{5}}{12} + \frac {5 B c^{3} d e^{4}}{12}\right ) + x^{11} \cdot \left (\frac {5 A c^{3} d e^{4}}{11} + \frac {3 B a c^{2} e^{5}}{11} + \frac {10 B c^{3} d^{2} e^{3}}{11}\right ) + x^{10} \cdot \left (\frac {3 A a c^{2} e^{5}}{10} + A c^{3} d^{2} e^{3} + \frac {3 B a c^{2} d e^{4}}{2} + B c^{3} d^{3} e^{2}\right ) + x^{9} \cdot \left (\frac {5 A a c^{2} d e^{4}}{3} + \frac {10 A c^{3} d^{3} e^{2}}{9} + \frac {B a^{2} c e^{5}}{3} + \frac {10 B a c^{2} d^{2} e^{3}}{3} + \frac {5 B c^{3} d^{4} e}{9}\right ) + x^{8} \cdot \left (\frac {3 A a^{2} c e^{5}}{8} + \frac {15 A a c^{2} d^{2} e^{3}}{4} + \frac {5 A c^{3} d^{4} e}{8} + \frac {15 B a^{2} c d e^{4}}{8} + \frac {15 B a c^{2} d^{3} e^{2}}{4} + \frac {B c^{3} d^{5}}{8}\right ) + x^{7} \cdot \left (\frac {15 A a^{2} c d e^{4}}{7} + \frac {30 A a c^{2} d^{3} e^{2}}{7} + \frac {A c^{3} d^{5}}{7} + \frac {B a^{3} e^{5}}{7} + \frac {30 B a^{2} c d^{2} e^{3}}{7} + \frac {15 B a c^{2} d^{4} e}{7}\right ) + x^{6} \left (\frac {A a^{3} e^{5}}{6} + 5 A a^{2} c d^{2} e^{3} + \frac {5 A a c^{2} d^{4} e}{2} + \frac {5 B a^{3} d e^{4}}{6} + 5 B a^{2} c d^{3} e^{2} + \frac {B a c^{2} d^{5}}{2}\right ) + x^{5} \left (A a^{3} d e^{4} + 6 A a^{2} c d^{3} e^{2} + \frac {3 A a c^{2} d^{5}}{5} + 2 B a^{3} d^{2} e^{3} + 3 B a^{2} c d^{4} e\right ) + x^{4} \cdot \left (\frac {5 A a^{3} d^{2} e^{3}}{2} + \frac {15 A a^{2} c d^{4} e}{4} + \frac {5 B a^{3} d^{3} e^{2}}{2} + \frac {3 B a^{2} c d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {10 A a^{3} d^{3} e^{2}}{3} + A a^{2} c d^{5} + \frac {5 B a^{3} d^{4} e}{3}\right ) + x^{2} \cdot \left (\frac {5 A a^{3} d^{4} e}{2} + \frac {B a^{3} d^{5}}{2}\right ) \] Input:

integrate((B*x+A)*(e*x+d)**5*(c*x**2+a)**3,x)
 

Output:

A*a**3*d**5*x + B*c**3*e**5*x**13/13 + x**12*(A*c**3*e**5/12 + 5*B*c**3*d* 
e**4/12) + x**11*(5*A*c**3*d*e**4/11 + 3*B*a*c**2*e**5/11 + 10*B*c**3*d**2 
*e**3/11) + x**10*(3*A*a*c**2*e**5/10 + A*c**3*d**2*e**3 + 3*B*a*c**2*d*e* 
*4/2 + B*c**3*d**3*e**2) + x**9*(5*A*a*c**2*d*e**4/3 + 10*A*c**3*d**3*e**2 
/9 + B*a**2*c*e**5/3 + 10*B*a*c**2*d**2*e**3/3 + 5*B*c**3*d**4*e/9) + x**8 
*(3*A*a**2*c*e**5/8 + 15*A*a*c**2*d**2*e**3/4 + 5*A*c**3*d**4*e/8 + 15*B*a 
**2*c*d*e**4/8 + 15*B*a*c**2*d**3*e**2/4 + B*c**3*d**5/8) + x**7*(15*A*a** 
2*c*d*e**4/7 + 30*A*a*c**2*d**3*e**2/7 + A*c**3*d**5/7 + B*a**3*e**5/7 + 3 
0*B*a**2*c*d**2*e**3/7 + 15*B*a*c**2*d**4*e/7) + x**6*(A*a**3*e**5/6 + 5*A 
*a**2*c*d**2*e**3 + 5*A*a*c**2*d**4*e/2 + 5*B*a**3*d*e**4/6 + 5*B*a**2*c*d 
**3*e**2 + B*a*c**2*d**5/2) + x**5*(A*a**3*d*e**4 + 6*A*a**2*c*d**3*e**2 + 
 3*A*a*c**2*d**5/5 + 2*B*a**3*d**2*e**3 + 3*B*a**2*c*d**4*e) + x**4*(5*A*a 
**3*d**2*e**3/2 + 15*A*a**2*c*d**4*e/4 + 5*B*a**3*d**3*e**2/2 + 3*B*a**2*c 
*d**5/4) + x**3*(10*A*a**3*d**3*e**2/3 + A*a**2*c*d**5 + 5*B*a**3*d**4*e/3 
) + x**2*(5*A*a**3*d**4*e/2 + B*a**3*d**5/2)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.75 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right )^3 \, dx=\frac {1}{13} \, B c^{3} e^{5} x^{13} + \frac {1}{12} \, {\left (5 \, B c^{3} d e^{4} + A c^{3} e^{5}\right )} x^{12} + \frac {1}{11} \, {\left (10 \, B c^{3} d^{2} e^{3} + 5 \, A c^{3} d e^{4} + 3 \, B a c^{2} e^{5}\right )} x^{11} + \frac {1}{10} \, {\left (10 \, B c^{3} d^{3} e^{2} + 10 \, A c^{3} d^{2} e^{3} + 15 \, B a c^{2} d e^{4} + 3 \, A a c^{2} e^{5}\right )} x^{10} + A a^{3} d^{5} x + \frac {1}{9} \, {\left (5 \, B c^{3} d^{4} e + 10 \, A c^{3} d^{3} e^{2} + 30 \, B a c^{2} d^{2} e^{3} + 15 \, A a c^{2} d e^{4} + 3 \, B a^{2} c e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{5} + 5 \, A c^{3} d^{4} e + 30 \, B a c^{2} d^{3} e^{2} + 30 \, A a c^{2} d^{2} e^{3} + 15 \, B a^{2} c d e^{4} + 3 \, A a^{2} c e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d^{5} + 15 \, B a c^{2} d^{4} e + 30 \, A a c^{2} d^{3} e^{2} + 30 \, B a^{2} c d^{2} e^{3} + 15 \, A a^{2} c d e^{4} + B a^{3} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, B a c^{2} d^{5} + 15 \, A a c^{2} d^{4} e + 30 \, B a^{2} c d^{3} e^{2} + 30 \, A a^{2} c d^{2} e^{3} + 5 \, B a^{3} d e^{4} + A a^{3} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A a c^{2} d^{5} + 15 \, B a^{2} c d^{4} e + 30 \, A a^{2} c d^{3} e^{2} + 10 \, B a^{3} d^{2} e^{3} + 5 \, A a^{3} d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{5} + 15 \, A a^{2} c d^{4} e + 10 \, B a^{3} d^{3} e^{2} + 10 \, A a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d^{5} + 5 \, B a^{3} d^{4} e + 10 \, A a^{3} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{5} + 5 \, A a^{3} d^{4} e\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a)^3,x, algorithm="maxima")
 

Output:

1/13*B*c^3*e^5*x^13 + 1/12*(5*B*c^3*d*e^4 + A*c^3*e^5)*x^12 + 1/11*(10*B*c 
^3*d^2*e^3 + 5*A*c^3*d*e^4 + 3*B*a*c^2*e^5)*x^11 + 1/10*(10*B*c^3*d^3*e^2 
+ 10*A*c^3*d^2*e^3 + 15*B*a*c^2*d*e^4 + 3*A*a*c^2*e^5)*x^10 + A*a^3*d^5*x 
+ 1/9*(5*B*c^3*d^4*e + 10*A*c^3*d^3*e^2 + 30*B*a*c^2*d^2*e^3 + 15*A*a*c^2* 
d*e^4 + 3*B*a^2*c*e^5)*x^9 + 1/8*(B*c^3*d^5 + 5*A*c^3*d^4*e + 30*B*a*c^2*d 
^3*e^2 + 30*A*a*c^2*d^2*e^3 + 15*B*a^2*c*d*e^4 + 3*A*a^2*c*e^5)*x^8 + 1/7* 
(A*c^3*d^5 + 15*B*a*c^2*d^4*e + 30*A*a*c^2*d^3*e^2 + 30*B*a^2*c*d^2*e^3 + 
15*A*a^2*c*d*e^4 + B*a^3*e^5)*x^7 + 1/6*(3*B*a*c^2*d^5 + 15*A*a*c^2*d^4*e 
+ 30*B*a^2*c*d^3*e^2 + 30*A*a^2*c*d^2*e^3 + 5*B*a^3*d*e^4 + A*a^3*e^5)*x^6 
 + 1/5*(3*A*a*c^2*d^5 + 15*B*a^2*c*d^4*e + 30*A*a^2*c*d^3*e^2 + 10*B*a^3*d 
^2*e^3 + 5*A*a^3*d*e^4)*x^5 + 1/4*(3*B*a^2*c*d^5 + 15*A*a^2*c*d^4*e + 10*B 
*a^3*d^3*e^2 + 10*A*a^3*d^2*e^3)*x^4 + 1/3*(3*A*a^2*c*d^5 + 5*B*a^3*d^4*e 
+ 10*A*a^3*d^3*e^2)*x^3 + 1/2*(B*a^3*d^5 + 5*A*a^3*d^4*e)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (318) = 636\).

Time = 0.12 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.97 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right )^3 \, dx=\frac {1}{13} \, B c^{3} e^{5} x^{13} + \frac {5}{12} \, B c^{3} d e^{4} x^{12} + \frac {1}{12} \, A c^{3} e^{5} x^{12} + \frac {10}{11} \, B c^{3} d^{2} e^{3} x^{11} + \frac {5}{11} \, A c^{3} d e^{4} x^{11} + \frac {3}{11} \, B a c^{2} e^{5} x^{11} + B c^{3} d^{3} e^{2} x^{10} + A c^{3} d^{2} e^{3} x^{10} + \frac {3}{2} \, B a c^{2} d e^{4} x^{10} + \frac {3}{10} \, A a c^{2} e^{5} x^{10} + \frac {5}{9} \, B c^{3} d^{4} e x^{9} + \frac {10}{9} \, A c^{3} d^{3} e^{2} x^{9} + \frac {10}{3} \, B a c^{2} d^{2} e^{3} x^{9} + \frac {5}{3} \, A a c^{2} d e^{4} x^{9} + \frac {1}{3} \, B a^{2} c e^{5} x^{9} + \frac {1}{8} \, B c^{3} d^{5} x^{8} + \frac {5}{8} \, A c^{3} d^{4} e x^{8} + \frac {15}{4} \, B a c^{2} d^{3} e^{2} x^{8} + \frac {15}{4} \, A a c^{2} d^{2} e^{3} x^{8} + \frac {15}{8} \, B a^{2} c d e^{4} x^{8} + \frac {3}{8} \, A a^{2} c e^{5} x^{8} + \frac {1}{7} \, A c^{3} d^{5} x^{7} + \frac {15}{7} \, B a c^{2} d^{4} e x^{7} + \frac {30}{7} \, A a c^{2} d^{3} e^{2} x^{7} + \frac {30}{7} \, B a^{2} c d^{2} e^{3} x^{7} + \frac {15}{7} \, A a^{2} c d e^{4} x^{7} + \frac {1}{7} \, B a^{3} e^{5} x^{7} + \frac {1}{2} \, B a c^{2} d^{5} x^{6} + \frac {5}{2} \, A a c^{2} d^{4} e x^{6} + 5 \, B a^{2} c d^{3} e^{2} x^{6} + 5 \, A a^{2} c d^{2} e^{3} x^{6} + \frac {5}{6} \, B a^{3} d e^{4} x^{6} + \frac {1}{6} \, A a^{3} e^{5} x^{6} + \frac {3}{5} \, A a c^{2} d^{5} x^{5} + 3 \, B a^{2} c d^{4} e x^{5} + 6 \, A a^{2} c d^{3} e^{2} x^{5} + 2 \, B a^{3} d^{2} e^{3} x^{5} + A a^{3} d e^{4} x^{5} + \frac {3}{4} \, B a^{2} c d^{5} x^{4} + \frac {15}{4} \, A a^{2} c d^{4} e x^{4} + \frac {5}{2} \, B a^{3} d^{3} e^{2} x^{4} + \frac {5}{2} \, A a^{3} d^{2} e^{3} x^{4} + A a^{2} c d^{5} x^{3} + \frac {5}{3} \, B a^{3} d^{4} e x^{3} + \frac {10}{3} \, A a^{3} d^{3} e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{5} x^{2} + \frac {5}{2} \, A a^{3} d^{4} e x^{2} + A a^{3} d^{5} x \] Input:

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a)^3,x, algorithm="giac")
 

Output:

1/13*B*c^3*e^5*x^13 + 5/12*B*c^3*d*e^4*x^12 + 1/12*A*c^3*e^5*x^12 + 10/11* 
B*c^3*d^2*e^3*x^11 + 5/11*A*c^3*d*e^4*x^11 + 3/11*B*a*c^2*e^5*x^11 + B*c^3 
*d^3*e^2*x^10 + A*c^3*d^2*e^3*x^10 + 3/2*B*a*c^2*d*e^4*x^10 + 3/10*A*a*c^2 
*e^5*x^10 + 5/9*B*c^3*d^4*e*x^9 + 10/9*A*c^3*d^3*e^2*x^9 + 10/3*B*a*c^2*d^ 
2*e^3*x^9 + 5/3*A*a*c^2*d*e^4*x^9 + 1/3*B*a^2*c*e^5*x^9 + 1/8*B*c^3*d^5*x^ 
8 + 5/8*A*c^3*d^4*e*x^8 + 15/4*B*a*c^2*d^3*e^2*x^8 + 15/4*A*a*c^2*d^2*e^3* 
x^8 + 15/8*B*a^2*c*d*e^4*x^8 + 3/8*A*a^2*c*e^5*x^8 + 1/7*A*c^3*d^5*x^7 + 1 
5/7*B*a*c^2*d^4*e*x^7 + 30/7*A*a*c^2*d^3*e^2*x^7 + 30/7*B*a^2*c*d^2*e^3*x^ 
7 + 15/7*A*a^2*c*d*e^4*x^7 + 1/7*B*a^3*e^5*x^7 + 1/2*B*a*c^2*d^5*x^6 + 5/2 
*A*a*c^2*d^4*e*x^6 + 5*B*a^2*c*d^3*e^2*x^6 + 5*A*a^2*c*d^2*e^3*x^6 + 5/6*B 
*a^3*d*e^4*x^6 + 1/6*A*a^3*e^5*x^6 + 3/5*A*a*c^2*d^5*x^5 + 3*B*a^2*c*d^4*e 
*x^5 + 6*A*a^2*c*d^3*e^2*x^5 + 2*B*a^3*d^2*e^3*x^5 + A*a^3*d*e^4*x^5 + 3/4 
*B*a^2*c*d^5*x^4 + 15/4*A*a^2*c*d^4*e*x^4 + 5/2*B*a^3*d^3*e^2*x^4 + 5/2*A* 
a^3*d^2*e^3*x^4 + A*a^2*c*d^5*x^3 + 5/3*B*a^3*d^4*e*x^3 + 10/3*A*a^3*d^3*e 
^2*x^3 + 1/2*B*a^3*d^5*x^2 + 5/2*A*a^3*d^4*e*x^2 + A*a^3*d^5*x
 

Mupad [B] (verification not implemented)

Time = 6.84 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.62 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right )^3 \, dx=x^6\,\left (\frac {5\,B\,a^3\,d\,e^4}{6}+\frac {A\,a^3\,e^5}{6}+5\,B\,a^2\,c\,d^3\,e^2+5\,A\,a^2\,c\,d^2\,e^3+\frac {B\,a\,c^2\,d^5}{2}+\frac {5\,A\,a\,c^2\,d^4\,e}{2}\right )+x^7\,\left (\frac {B\,a^3\,e^5}{7}+\frac {30\,B\,a^2\,c\,d^2\,e^3}{7}+\frac {15\,A\,a^2\,c\,d\,e^4}{7}+\frac {15\,B\,a\,c^2\,d^4\,e}{7}+\frac {30\,A\,a\,c^2\,d^3\,e^2}{7}+\frac {A\,c^3\,d^5}{7}\right )+x^8\,\left (\frac {15\,B\,a^2\,c\,d\,e^4}{8}+\frac {3\,A\,a^2\,c\,e^5}{8}+\frac {15\,B\,a\,c^2\,d^3\,e^2}{4}+\frac {15\,A\,a\,c^2\,d^2\,e^3}{4}+\frac {B\,c^3\,d^5}{8}+\frac {5\,A\,c^3\,d^4\,e}{8}\right )+x^5\,\left (2\,B\,a^3\,d^2\,e^3+A\,a^3\,d\,e^4+3\,B\,a^2\,c\,d^4\,e+6\,A\,a^2\,c\,d^3\,e^2+\frac {3\,A\,a\,c^2\,d^5}{5}\right )+x^9\,\left (\frac {B\,a^2\,c\,e^5}{3}+\frac {10\,B\,a\,c^2\,d^2\,e^3}{3}+\frac {5\,A\,a\,c^2\,d\,e^4}{3}+\frac {5\,B\,c^3\,d^4\,e}{9}+\frac {10\,A\,c^3\,d^3\,e^2}{9}\right )+\frac {c^2\,e^2\,x^{10}\,\left (10\,B\,c\,d^3+10\,A\,c\,d^2\,e+15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{10}+\frac {a^3\,d^4\,x^2\,\left (5\,A\,e+B\,d\right )}{2}+\frac {c^3\,e^4\,x^{12}\,\left (A\,e+5\,B\,d\right )}{12}+\frac {a^2\,d^3\,x^3\,\left (3\,A\,c\,d^2+5\,B\,a\,d\,e+10\,A\,a\,e^2\right )}{3}+\frac {c^2\,e^3\,x^{11}\,\left (10\,B\,c\,d^2+5\,A\,c\,d\,e+3\,B\,a\,e^2\right )}{11}+A\,a^3\,d^5\,x+\frac {a^2\,d^2\,x^4\,\left (3\,B\,c\,d^3+15\,A\,c\,d^2\,e+10\,B\,a\,d\,e^2+10\,A\,a\,e^3\right )}{4}+\frac {B\,c^3\,e^5\,x^{13}}{13} \] Input:

int((a + c*x^2)^3*(A + B*x)*(d + e*x)^5,x)
 

Output:

x^6*((A*a^3*e^5)/6 + (B*a*c^2*d^5)/2 + (5*B*a^3*d*e^4)/6 + 5*A*a^2*c*d^2*e 
^3 + 5*B*a^2*c*d^3*e^2 + (5*A*a*c^2*d^4*e)/2) + x^7*((A*c^3*d^5)/7 + (B*a^ 
3*e^5)/7 + (30*A*a*c^2*d^3*e^2)/7 + (30*B*a^2*c*d^2*e^3)/7 + (15*A*a^2*c*d 
*e^4)/7 + (15*B*a*c^2*d^4*e)/7) + x^8*((B*c^3*d^5)/8 + (3*A*a^2*c*e^5)/8 + 
 (5*A*c^3*d^4*e)/8 + (15*A*a*c^2*d^2*e^3)/4 + (15*B*a*c^2*d^3*e^2)/4 + (15 
*B*a^2*c*d*e^4)/8) + x^5*((3*A*a*c^2*d^5)/5 + A*a^3*d*e^4 + 2*B*a^3*d^2*e^ 
3 + 6*A*a^2*c*d^3*e^2 + 3*B*a^2*c*d^4*e) + x^9*((B*a^2*c*e^5)/3 + (5*B*c^3 
*d^4*e)/9 + (10*A*c^3*d^3*e^2)/9 + (10*B*a*c^2*d^2*e^3)/3 + (5*A*a*c^2*d*e 
^4)/3) + (c^2*e^2*x^10*(3*A*a*e^3 + 10*B*c*d^3 + 15*B*a*d*e^2 + 10*A*c*d^2 
*e))/10 + (a^3*d^4*x^2*(5*A*e + B*d))/2 + (c^3*e^4*x^12*(A*e + 5*B*d))/12 
+ (a^2*d^3*x^3*(10*A*a*e^2 + 3*A*c*d^2 + 5*B*a*d*e))/3 + (c^2*e^3*x^11*(3* 
B*a*e^2 + 10*B*c*d^2 + 5*A*c*d*e))/11 + A*a^3*d^5*x + (a^2*d^2*x^4*(10*A*a 
*e^3 + 3*B*c*d^3 + 10*B*a*d*e^2 + 15*A*c*d^2*e))/4 + (B*c^3*e^5*x^13)/13
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.96 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right )^3 \, dx=\frac {x \left (27720 b \,c^{3} e^{5} x^{12}+30030 a \,c^{3} e^{5} x^{11}+150150 b \,c^{3} d \,e^{4} x^{11}+98280 a b \,c^{2} e^{5} x^{10}+163800 a \,c^{3} d \,e^{4} x^{10}+327600 b \,c^{3} d^{2} e^{3} x^{10}+108108 a^{2} c^{2} e^{5} x^{9}+540540 a b \,c^{2} d \,e^{4} x^{9}+360360 a \,c^{3} d^{2} e^{3} x^{9}+360360 b \,c^{3} d^{3} e^{2} x^{9}+120120 a^{2} b c \,e^{5} x^{8}+600600 a^{2} c^{2} d \,e^{4} x^{8}+1201200 a b \,c^{2} d^{2} e^{3} x^{8}+400400 a \,c^{3} d^{3} e^{2} x^{8}+200200 b \,c^{3} d^{4} e \,x^{8}+135135 a^{3} c \,e^{5} x^{7}+675675 a^{2} b c d \,e^{4} x^{7}+1351350 a^{2} c^{2} d^{2} e^{3} x^{7}+1351350 a b \,c^{2} d^{3} e^{2} x^{7}+225225 a \,c^{3} d^{4} e \,x^{7}+45045 b \,c^{3} d^{5} x^{7}+51480 a^{3} b \,e^{5} x^{6}+772200 a^{3} c d \,e^{4} x^{6}+1544400 a^{2} b c \,d^{2} e^{3} x^{6}+1544400 a^{2} c^{2} d^{3} e^{2} x^{6}+772200 a b \,c^{2} d^{4} e \,x^{6}+51480 a \,c^{3} d^{5} x^{6}+60060 a^{4} e^{5} x^{5}+300300 a^{3} b d \,e^{4} x^{5}+1801800 a^{3} c \,d^{2} e^{3} x^{5}+1801800 a^{2} b c \,d^{3} e^{2} x^{5}+900900 a^{2} c^{2} d^{4} e \,x^{5}+180180 a b \,c^{2} d^{5} x^{5}+360360 a^{4} d \,e^{4} x^{4}+720720 a^{3} b \,d^{2} e^{3} x^{4}+2162160 a^{3} c \,d^{3} e^{2} x^{4}+1081080 a^{2} b c \,d^{4} e \,x^{4}+216216 a^{2} c^{2} d^{5} x^{4}+900900 a^{4} d^{2} e^{3} x^{3}+900900 a^{3} b \,d^{3} e^{2} x^{3}+1351350 a^{3} c \,d^{4} e \,x^{3}+270270 a^{2} b c \,d^{5} x^{3}+1201200 a^{4} d^{3} e^{2} x^{2}+600600 a^{3} b \,d^{4} e \,x^{2}+360360 a^{3} c \,d^{5} x^{2}+900900 a^{4} d^{4} e x +180180 a^{3} b \,d^{5} x +360360 a^{4} d^{5}\right )}{360360} \] Input:

int((B*x+A)*(e*x+d)^5*(c*x^2+a)^3,x)
 

Output:

(x*(360360*a**4*d**5 + 900900*a**4*d**4*e*x + 1201200*a**4*d**3*e**2*x**2 
+ 900900*a**4*d**2*e**3*x**3 + 360360*a**4*d*e**4*x**4 + 60060*a**4*e**5*x 
**5 + 180180*a**3*b*d**5*x + 600600*a**3*b*d**4*e*x**2 + 900900*a**3*b*d** 
3*e**2*x**3 + 720720*a**3*b*d**2*e**3*x**4 + 300300*a**3*b*d*e**4*x**5 + 5 
1480*a**3*b*e**5*x**6 + 360360*a**3*c*d**5*x**2 + 1351350*a**3*c*d**4*e*x* 
*3 + 2162160*a**3*c*d**3*e**2*x**4 + 1801800*a**3*c*d**2*e**3*x**5 + 77220 
0*a**3*c*d*e**4*x**6 + 135135*a**3*c*e**5*x**7 + 270270*a**2*b*c*d**5*x**3 
 + 1081080*a**2*b*c*d**4*e*x**4 + 1801800*a**2*b*c*d**3*e**2*x**5 + 154440 
0*a**2*b*c*d**2*e**3*x**6 + 675675*a**2*b*c*d*e**4*x**7 + 120120*a**2*b*c* 
e**5*x**8 + 216216*a**2*c**2*d**5*x**4 + 900900*a**2*c**2*d**4*e*x**5 + 15 
44400*a**2*c**2*d**3*e**2*x**6 + 1351350*a**2*c**2*d**2*e**3*x**7 + 600600 
*a**2*c**2*d*e**4*x**8 + 108108*a**2*c**2*e**5*x**9 + 180180*a*b*c**2*d**5 
*x**5 + 772200*a*b*c**2*d**4*e*x**6 + 1351350*a*b*c**2*d**3*e**2*x**7 + 12 
01200*a*b*c**2*d**2*e**3*x**8 + 540540*a*b*c**2*d*e**4*x**9 + 98280*a*b*c* 
*2*e**5*x**10 + 51480*a*c**3*d**5*x**6 + 225225*a*c**3*d**4*e*x**7 + 40040 
0*a*c**3*d**3*e**2*x**8 + 360360*a*c**3*d**2*e**3*x**9 + 163800*a*c**3*d*e 
**4*x**10 + 30030*a*c**3*e**5*x**11 + 45045*b*c**3*d**5*x**7 + 200200*b*c* 
*3*d**4*e*x**8 + 360360*b*c**3*d**3*e**2*x**9 + 327600*b*c**3*d**2*e**3*x* 
*10 + 150150*b*c**3*d*e**4*x**11 + 27720*b*c**3*e**5*x**12))/360360