\(\int (A+B x) (d+e x)^2 (a+c x^2)^3 \, dx\) [65]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.71 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^3 \, dx=a^3 A d^2 x+\frac {1}{2} a^3 d (B d+2 A e) x^2+\frac {1}{3} a^2 \left (3 A c d^2+2 a B d e+a A e^2\right ) x^3+\frac {1}{4} a^2 \left (3 B c d^2+6 A c d e+a B e^2\right ) x^4+\frac {3}{5} a c \left (A c d^2+2 a B d e+a A e^2\right ) x^5+\frac {1}{2} a c \left (B c d^2+2 A c d e+a B e^2\right ) x^6+\frac {1}{7} c^2 \left (A c d^2+6 a B d e+3 a A e^2\right ) x^7+\frac {1}{8} c^2 \left (B c d^2+2 A c d e+3 a B e^2\right ) x^8+\frac {1}{9} c^3 e (2 B d+A e) x^9+\frac {1}{10} B c^3 e^2 x^{10} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.62 (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.

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

\(\Big \downarrow \) 652

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/3*((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^3)/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)^4)/(4*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)^5)/(5*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)^6)/(6*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)^7)/(7*e^8) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + 
a*B*e^2)*(d + e*x)^8)/(8*e^8) - (c^3*(7*B*d - A*e)*(d + e*x)^9)/(9*e^8) + 
(B*c^3*(d + e*x)^10)/(10*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 = 251, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {B \,e^{2} c^{3} x^{10}}{10}+\frac {\left (A \,e^{2}+2 B d e \right ) c^{3} x^{9}}{9}+\frac {\left (\left (2 A d e +B \,d^{2}\right ) c^{3}+3 B \,e^{2} a \,c^{2}\right ) x^{8}}{8}+\frac {\left (A \,c^{3} d^{2}+3 \left (A \,e^{2}+2 B d e \right ) a \,c^{2}\right ) x^{7}}{7}+\frac {\left (3 \left (2 A d e +B \,d^{2}\right ) a \,c^{2}+3 B \,e^{2} a^{2} c \right ) x^{6}}{6}+\frac {\left (3 A \,d^{2} a \,c^{2}+3 \left (A \,e^{2}+2 B d e \right ) a^{2} c \right ) x^{5}}{5}+\frac {\left (3 \left (2 A d e +B \,d^{2}\right ) a^{2} c +B \,e^{2} a^{3}\right ) x^{4}}{4}+\frac {\left (3 A \,d^{2} a^{2} c +\left (A \,e^{2}+2 B d e \right ) a^{3}\right ) x^{3}}{3}+\frac {\left (2 A d e +B \,d^{2}\right ) a^{3} x^{2}}{2}+A \,d^{2} a^{3} x\)	\(251\)
norman	\(\frac {B \,e^{2} c^{3} x^{10}}{10}+\left (\frac {1}{9} A \,c^{3} e^{2}+\frac {2}{9} B \,c^{3} d e \right ) x^{9}+\left (\frac {1}{4} A \,c^{3} d e +\frac {3}{8} B \,e^{2} a \,c^{2}+\frac {1}{8} B \,c^{3} d^{2}\right ) x^{8}+\left (\frac {3}{7} A a \,c^{2} e^{2}+\frac {1}{7} A \,c^{3} d^{2}+\frac {6}{7} B a \,c^{2} d e \right ) x^{7}+\left (A a \,c^{2} d e +\frac {1}{2} B \,e^{2} a^{2} c +\frac {1}{2} B a \,c^{2} d^{2}\right ) x^{6}+\left (\frac {3}{5} A \,a^{2} c \,e^{2}+\frac {3}{5} A \,d^{2} a \,c^{2}+\frac {6}{5} B \,a^{2} c d e \right ) x^{5}+\left (\frac {3}{2} A \,a^{2} c d e +\frac {1}{4} B \,e^{2} a^{3}+\frac {3}{4} B \,a^{2} c \,d^{2}\right ) x^{4}+\left (\frac {1}{3} A \,a^{3} e^{2}+A \,d^{2} a^{2} c +\frac {2}{3} B \,a^{3} d e \right ) x^{3}+\left (A \,a^{3} d e +\frac {1}{2} B \,a^{3} d^{2}\right ) x^{2}+A \,d^{2} a^{3} x\)	\(262\)
gosper	\(\frac {1}{10} B \,e^{2} c^{3} x^{10}+\frac {1}{9} x^{9} A \,c^{3} e^{2}+\frac {2}{9} x^{9} B \,c^{3} d e +\frac {1}{4} x^{8} A \,c^{3} d e +\frac {3}{8} x^{8} B \,e^{2} a \,c^{2}+\frac {1}{8} x^{8} B \,c^{3} d^{2}+\frac {3}{7} x^{7} A a \,c^{2} e^{2}+\frac {1}{7} x^{7} A \,c^{3} d^{2}+\frac {6}{7} x^{7} B a \,c^{2} d e +x^{6} A a \,c^{2} d e +\frac {1}{2} x^{6} B \,e^{2} a^{2} c +\frac {1}{2} x^{6} B a \,c^{2} d^{2}+\frac {3}{5} x^{5} A \,a^{2} c \,e^{2}+\frac {3}{5} x^{5} A \,d^{2} a \,c^{2}+\frac {6}{5} x^{5} B \,a^{2} c d e +\frac {3}{2} x^{4} A \,a^{2} c d e +\frac {1}{4} x^{4} B \,e^{2} a^{3}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{2}+\frac {1}{3} x^{3} A \,a^{3} e^{2}+x^{3} A \,d^{2} a^{2} c +\frac {2}{3} x^{3} B \,a^{3} d e +x^{2} A \,a^{3} d e +\frac {1}{2} x^{2} B \,a^{3} d^{2}+A \,d^{2} a^{3} x\)	\(288\)
risch	\(\frac {1}{10} B \,e^{2} c^{3} x^{10}+\frac {1}{9} x^{9} A \,c^{3} e^{2}+\frac {2}{9} x^{9} B \,c^{3} d e +\frac {1}{4} x^{8} A \,c^{3} d e +\frac {3}{8} x^{8} B \,e^{2} a \,c^{2}+\frac {1}{8} x^{8} B \,c^{3} d^{2}+\frac {3}{7} x^{7} A a \,c^{2} e^{2}+\frac {1}{7} x^{7} A \,c^{3} d^{2}+\frac {6}{7} x^{7} B a \,c^{2} d e +x^{6} A a \,c^{2} d e +\frac {1}{2} x^{6} B \,e^{2} a^{2} c +\frac {1}{2} x^{6} B a \,c^{2} d^{2}+\frac {3}{5} x^{5} A \,a^{2} c \,e^{2}+\frac {3}{5} x^{5} A \,d^{2} a \,c^{2}+\frac {6}{5} x^{5} B \,a^{2} c d e +\frac {3}{2} x^{4} A \,a^{2} c d e +\frac {1}{4} x^{4} B \,e^{2} a^{3}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{2}+\frac {1}{3} x^{3} A \,a^{3} e^{2}+x^{3} A \,d^{2} a^{2} c +\frac {2}{3} x^{3} B \,a^{3} d e +x^{2} A \,a^{3} d e +\frac {1}{2} x^{2} B \,a^{3} d^{2}+A \,d^{2} a^{3} x\)	\(288\)
parallelrisch	\(\frac {1}{10} B \,e^{2} c^{3} x^{10}+\frac {1}{9} x^{9} A \,c^{3} e^{2}+\frac {2}{9} x^{9} B \,c^{3} d e +\frac {1}{4} x^{8} A \,c^{3} d e +\frac {3}{8} x^{8} B \,e^{2} a \,c^{2}+\frac {1}{8} x^{8} B \,c^{3} d^{2}+\frac {3}{7} x^{7} A a \,c^{2} e^{2}+\frac {1}{7} x^{7} A \,c^{3} d^{2}+\frac {6}{7} x^{7} B a \,c^{2} d e +x^{6} A a \,c^{2} d e +\frac {1}{2} x^{6} B \,e^{2} a^{2} c +\frac {1}{2} x^{6} B a \,c^{2} d^{2}+\frac {3}{5} x^{5} A \,a^{2} c \,e^{2}+\frac {3}{5} x^{5} A \,d^{2} a \,c^{2}+\frac {6}{5} x^{5} B \,a^{2} c d e +\frac {3}{2} x^{4} A \,a^{2} c d e +\frac {1}{4} x^{4} B \,e^{2} a^{3}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{2}+\frac {1}{3} x^{3} A \,a^{3} e^{2}+x^{3} A \,d^{2} a^{2} c +\frac {2}{3} x^{3} B \,a^{3} d e +x^{2} A \,a^{3} d e +\frac {1}{2} x^{2} B \,a^{3} d^{2}+A \,d^{2} a^{3} x\)	\(288\)
orering	\(\frac {x \left (252 B \,e^{2} c^{3} x^{9}+280 A \,c^{3} e^{2} x^{8}+560 B \,c^{3} d e \,x^{8}+630 A \,c^{3} d e \,x^{7}+945 B a \,c^{2} e^{2} x^{7}+315 B \,c^{3} d^{2} x^{7}+1080 A a \,c^{2} e^{2} x^{6}+360 A \,c^{3} d^{2} x^{6}+2160 B a \,c^{2} d e \,x^{6}+2520 A a \,c^{2} d e \,x^{5}+1260 B \,a^{2} c \,e^{2} x^{5}+1260 B a \,c^{2} d^{2} x^{5}+1512 A \,a^{2} c \,e^{2} x^{4}+1512 A a \,c^{2} d^{2} x^{4}+3024 B \,a^{2} c d e \,x^{4}+3780 A \,a^{2} c d e \,x^{3}+630 B \,a^{3} e^{2} x^{3}+1890 B \,a^{2} c \,d^{2} x^{3}+840 A \,a^{3} e^{2} x^{2}+2520 A \,a^{2} c \,d^{2} x^{2}+1680 B \,a^{3} d e \,x^{2}+2520 A \,a^{3} d e x +1260 B \,a^{3} d^{2} x +2520 A \,d^{2} a^{3}\right )}{2520}\)	\(290\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.78 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^3 \, dx=\frac {1}{10} \, B c^{3} e^{2} x^{10} + \frac {1}{9} \, {\left (2 \, B c^{3} d e + A c^{3} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{2} + 2 \, A c^{3} d e + 3 \, B a c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d^{2} + 6 \, B a c^{2} d e + 3 \, A a c^{2} e^{2}\right )} x^{7} + A a^{3} d^{2} x + \frac {1}{2} \, {\left (B a c^{2} d^{2} + 2 \, A a c^{2} d e + B a^{2} c e^{2}\right )} x^{6} + \frac {3}{5} \, {\left (A a c^{2} d^{2} + 2 \, B a^{2} c d e + A a^{2} c e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{2} + 6 \, A a^{2} c d e + B a^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d^{2} + 2 \, B a^{3} d e + A a^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{2} + 2 \, A a^{3} d e\right )} x^{2} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.92 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^3 \, dx=A a^{3} d^{2} x + \frac {B c^{3} e^{2} x^{10}}{10} + x^{9} \left (\frac {A c^{3} e^{2}}{9} + \frac {2 B c^{3} d e}{9}\right ) + x^{8} \left (\frac {A c^{3} d e}{4} + \frac {3 B a c^{2} e^{2}}{8} + \frac {B c^{3} d^{2}}{8}\right ) + x^{7} \cdot \left (\frac {3 A a c^{2} e^{2}}{7} + \frac {A c^{3} d^{2}}{7} + \frac {6 B a c^{2} d e}{7}\right ) + x^{6} \left (A a c^{2} d e + \frac {B a^{2} c e^{2}}{2} + \frac {B a c^{2} d^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 A a^{2} c e^{2}}{5} + \frac {3 A a c^{2} d^{2}}{5} + \frac {6 B a^{2} c d e}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} c d e}{2} + \frac {B a^{3} e^{2}}{4} + \frac {3 B a^{2} c d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{3} e^{2}}{3} + A a^{2} c d^{2} + \frac {2 B a^{3} d e}{3}\right ) + x^{2} \left (A a^{3} d e + \frac {B a^{3} d^{2}}{2}\right ) \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.78 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^3 \, dx=\frac {1}{10} \, B c^{3} e^{2} x^{10} + \frac {1}{9} \, {\left (2 \, B c^{3} d e + A c^{3} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{2} + 2 \, A c^{3} d e + 3 \, B a c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d^{2} + 6 \, B a c^{2} d e + 3 \, A a c^{2} e^{2}\right )} x^{7} + A a^{3} d^{2} x + \frac {1}{2} \, {\left (B a c^{2} d^{2} + 2 \, A a c^{2} d e + B a^{2} c e^{2}\right )} x^{6} + \frac {3}{5} \, {\left (A a c^{2} d^{2} + 2 \, B a^{2} c d e + A a^{2} c e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{2} + 6 \, A a^{2} c d e + B a^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d^{2} + 2 \, B a^{3} d e + A a^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{2} + 2 \, A a^{3} d e\right )} x^{2} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.86 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^3 \, dx=\frac {1}{10} \, B c^{3} e^{2} x^{10} + \frac {2}{9} \, B c^{3} d e x^{9} + \frac {1}{9} \, A c^{3} e^{2} x^{9} + \frac {1}{8} \, B c^{3} d^{2} x^{8} + \frac {1}{4} \, A c^{3} d e x^{8} + \frac {3}{8} \, B a c^{2} e^{2} x^{8} + \frac {1}{7} \, A c^{3} d^{2} x^{7} + \frac {6}{7} \, B a c^{2} d e x^{7} + \frac {3}{7} \, A a c^{2} e^{2} x^{7} + \frac {1}{2} \, B a c^{2} d^{2} x^{6} + A a c^{2} d e x^{6} + \frac {1}{2} \, B a^{2} c e^{2} x^{6} + \frac {3}{5} \, A a c^{2} d^{2} x^{5} + \frac {6}{5} \, B a^{2} c d e x^{5} + \frac {3}{5} \, A a^{2} c e^{2} x^{5} + \frac {3}{4} \, B a^{2} c d^{2} x^{4} + \frac {3}{2} \, A a^{2} c d e x^{4} + \frac {1}{4} \, B a^{3} e^{2} x^{4} + A a^{2} c d^{2} x^{3} + \frac {2}{3} \, B a^{3} d e x^{3} + \frac {1}{3} \, A a^{3} e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{2} x^{2} + A a^{3} d e x^{2} + A a^{3} d^{2} x \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 6.89 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.68 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^3 \, dx=x^3\,\left (\frac {2\,B\,a^3\,d\,e}{3}+\frac {A\,a^3\,e^2}{3}+A\,c\,a^2\,d^2\right )+x^8\,\left (\frac {B\,c^3\,d^2}{8}+\frac {A\,c^3\,d\,e}{4}+\frac {3\,B\,a\,c^2\,e^2}{8}\right )+\frac {c^2\,x^7\,\left (A\,c\,d^2+6\,B\,a\,d\,e+3\,A\,a\,e^2\right )}{7}+\frac {a^2\,x^4\,\left (3\,B\,c\,d^2+6\,A\,c\,d\,e+B\,a\,e^2\right )}{4}+A\,a^3\,d^2\,x+\frac {3\,a\,c\,x^5\,\left (A\,c\,d^2+2\,B\,a\,d\,e+A\,a\,e^2\right )}{5}+\frac {a\,c\,x^6\,\left (B\,c\,d^2+2\,A\,c\,d\,e+B\,a\,e^2\right )}{2}+\frac {a^3\,d\,x^2\,\left (2\,A\,e+B\,d\right )}{2}+\frac {c^3\,e\,x^9\,\left (A\,e+2\,B\,d\right )}{9}+\frac {B\,c^3\,e^2\,x^{10}}{10} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.86 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^3 \, dx=\frac {x \left (252 b \,c^{3} e^{2} x^{9}+280 a \,c^{3} e^{2} x^{8}+560 b \,c^{3} d e \,x^{8}+945 a b \,c^{2} e^{2} x^{7}+630 a \,c^{3} d e \,x^{7}+315 b \,c^{3} d^{2} x^{7}+1080 a^{2} c^{2} e^{2} x^{6}+2160 a b \,c^{2} d e \,x^{6}+360 a \,c^{3} d^{2} x^{6}+1260 a^{2} b c \,e^{2} x^{5}+2520 a^{2} c^{2} d e \,x^{5}+1260 a b \,c^{2} d^{2} x^{5}+1512 a^{3} c \,e^{2} x^{4}+3024 a^{2} b c d e \,x^{4}+1512 a^{2} c^{2} d^{2} x^{4}+630 a^{3} b \,e^{2} x^{3}+3780 a^{3} c d e \,x^{3}+1890 a^{2} b c \,d^{2} x^{3}+840 a^{4} e^{2} x^{2}+1680 a^{3} b d e \,x^{2}+2520 a^{3} c \,d^{2} x^{2}+2520 a^{4} d e x +1260 a^{3} b \,d^{2} x +2520 a^{4} d^{2}\right )}{2520} \] Input:

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

Output:

(x*(2520*a**4*d**2 + 2520*a**4*d*e*x + 840*a**4*e**2*x**2 + 1260*a**3*b*d* 
*2*x + 1680*a**3*b*d*e*x**2 + 630*a**3*b*e**2*x**3 + 2520*a**3*c*d**2*x**2 
 + 3780*a**3*c*d*e*x**3 + 1512*a**3*c*e**2*x**4 + 1890*a**2*b*c*d**2*x**3 
+ 3024*a**2*b*c*d*e*x**4 + 1260*a**2*b*c*e**2*x**5 + 1512*a**2*c**2*d**2*x 
**4 + 2520*a**2*c**2*d*e*x**5 + 1080*a**2*c**2*e**2*x**6 + 1260*a*b*c**2*d 
**2*x**5 + 2160*a*b*c**2*d*e*x**6 + 945*a*b*c**2*e**2*x**7 + 360*a*c**3*d* 
*2*x**6 + 630*a*c**3*d*e*x**7 + 280*a*c**3*e**2*x**8 + 315*b*c**3*d**2*x** 
7 + 560*b*c**3*d*e*x**8 + 252*b*c**3*e**2*x**9))/2520