\(\int (A+B x) (d+e x)^2 (a+c x^2)^2 \, dx\) [49]

Optimal result

Integrand size = 22, antiderivative size = 206 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^2 \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^3}{3 e^6}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^4}{4 e^6}-\frac {2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^5}{5 e^6}+\frac {c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^6}{3 e^6}-\frac {c^2 (5 B d-A e) (d+e x)^7}{7 e^6}+\frac {B c^2 (d+e x)^8}{8 e^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.84 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^2 \, dx=a^2 A d^2 x+\frac {1}{2} a^2 d (B d+2 A e) x^2+\frac {1}{3} a \left (2 A c d^2+2 a B d e+a A e^2\right ) x^3+\frac {1}{4} a \left (2 B c d^2+4 A c d e+a B e^2\right ) x^4+\frac {1}{5} c \left (A c d^2+4 a B d e+2 a A e^2\right ) x^5+\frac {1}{6} c \left (B c d^2+2 A c d e+2 a B e^2\right ) x^6+\frac {1}{7} c^2 e (2 B d+A e) x^7+\frac {1}{8} B c^2 e^2 x^8 \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 206, 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.

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

\(\Big \downarrow \) 652

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 177, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {B \,e^{2} c^{2} x^{8}}{8}+\frac {\left (A \,e^{2}+2 B d e \right ) c^{2} x^{7}}{7}+\frac {\left (\left (2 A d e +B \,d^{2}\right ) c^{2}+2 B \,e^{2} a c \right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{2}+2 \left (A \,e^{2}+2 B d e \right ) a c \right ) x^{5}}{5}+\frac {\left (2 \left (2 A d e +B \,d^{2}\right ) a c +B \,e^{2} a^{2}\right ) x^{4}}{4}+\frac {\left (2 A \,d^{2} a c +\left (A \,e^{2}+2 B d e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 A d e +B \,d^{2}\right ) a^{2} x^{2}}{2}+A \,a^{2} d^{2} x\)	\(177\)
norman	\(\frac {B \,e^{2} c^{2} x^{8}}{8}+\left (\frac {1}{7} A \,c^{2} e^{2}+\frac {2}{7} B \,c^{2} d e \right ) x^{7}+\left (\frac {1}{3} A \,c^{2} d e +\frac {1}{3} B \,e^{2} a c +\frac {1}{6} B \,c^{2} d^{2}\right ) x^{6}+\left (\frac {2}{5} A a c \,e^{2}+\frac {1}{5} A \,c^{2} d^{2}+\frac {4}{5} B a c d e \right ) x^{5}+\left (A a c d e +\frac {1}{4} B \,e^{2} a^{2}+\frac {1}{2} a B c \,d^{2}\right ) x^{4}+\left (\frac {1}{3} A \,a^{2} e^{2}+\frac {2}{3} A \,d^{2} a c +\frac {2}{3} B \,a^{2} d e \right ) x^{3}+\left (A \,a^{2} d e +\frac {1}{2} a^{2} B \,d^{2}\right ) x^{2}+A \,a^{2} d^{2} x\)	\(183\)
gosper	\(\frac {1}{8} B \,e^{2} c^{2} x^{8}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B \,e^{2} a c +\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {2}{5} x^{5} A a c \,e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {4}{5} x^{5} B a c d e +x^{4} A a c d e +\frac {1}{4} x^{4} B \,e^{2} a^{2}+\frac {1}{2} x^{4} a B c \,d^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +x^{2} A \,a^{2} d e +\frac {1}{2} x^{2} a^{2} B \,d^{2}+A \,a^{2} d^{2} x\)	\(201\)
risch	\(\frac {1}{8} B \,e^{2} c^{2} x^{8}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B \,e^{2} a c +\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {2}{5} x^{5} A a c \,e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {4}{5} x^{5} B a c d e +x^{4} A a c d e +\frac {1}{4} x^{4} B \,e^{2} a^{2}+\frac {1}{2} x^{4} a B c \,d^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +x^{2} A \,a^{2} d e +\frac {1}{2} x^{2} a^{2} B \,d^{2}+A \,a^{2} d^{2} x\)	\(201\)
parallelrisch	\(\frac {1}{8} B \,e^{2} c^{2} x^{8}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B \,e^{2} a c +\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {2}{5} x^{5} A a c \,e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {4}{5} x^{5} B a c d e +x^{4} A a c d e +\frac {1}{4} x^{4} B \,e^{2} a^{2}+\frac {1}{2} x^{4} a B c \,d^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +x^{2} A \,a^{2} d e +\frac {1}{2} x^{2} a^{2} B \,d^{2}+A \,a^{2} d^{2} x\)	\(201\)
orering	\(\frac {x \left (105 B \,e^{2} c^{2} x^{7}+120 A \,c^{2} e^{2} x^{6}+240 B \,c^{2} d e \,x^{6}+280 A \,c^{2} d e \,x^{5}+280 B a c \,e^{2} x^{5}+140 B \,c^{2} d^{2} x^{5}+336 A a c \,e^{2} x^{4}+168 A \,c^{2} d^{2} x^{4}+672 B a c d e \,x^{4}+840 A a c d e \,x^{3}+210 B \,a^{2} e^{2} x^{3}+420 B a c \,d^{2} x^{3}+280 A \,a^{2} e^{2} x^{2}+560 A a c \,d^{2} x^{2}+560 B \,a^{2} d e \,x^{2}+840 A \,a^{2} d e x +420 B \,a^{2} d^{2} x +840 A \,d^{2} a^{2}\right )}{840}\)	\(202\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.89 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {1}{7} \, {\left (2 \, B c^{2} d e + A c^{2} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{2} + 2 \, A c^{2} d e + 2 \, B a c e^{2}\right )} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left (A c^{2} d^{2} + 4 \, B a c d e + 2 \, A a c e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, B a c d^{2} + 4 \, A a c d e + B a^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, A a c d^{2} + 2 \, B a^{2} d e + A a^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d^{2} + 2 \, A a^{2} d e\right )} x^{2} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.02 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^2 \, dx=A a^{2} d^{2} x + \frac {B c^{2} e^{2} x^{8}}{8} + x^{7} \left (\frac {A c^{2} e^{2}}{7} + \frac {2 B c^{2} d e}{7}\right ) + x^{6} \left (\frac {A c^{2} d e}{3} + \frac {B a c e^{2}}{3} + \frac {B c^{2} d^{2}}{6}\right ) + x^{5} \cdot \left (\frac {2 A a c e^{2}}{5} + \frac {A c^{2} d^{2}}{5} + \frac {4 B a c d e}{5}\right ) + x^{4} \left (A a c d e + \frac {B a^{2} e^{2}}{4} + \frac {B a c d^{2}}{2}\right ) + x^{3} \left (\frac {A a^{2} e^{2}}{3} + \frac {2 A a c d^{2}}{3} + \frac {2 B a^{2} d e}{3}\right ) + x^{2} \left (A a^{2} d e + \frac {B a^{2} d^{2}}{2}\right ) \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.89 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {1}{7} \, {\left (2 \, B c^{2} d e + A c^{2} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{2} + 2 \, A c^{2} d e + 2 \, B a c e^{2}\right )} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left (A c^{2} d^{2} + 4 \, B a c d e + 2 \, A a c e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, B a c d^{2} + 4 \, A a c d e + B a^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, A a c d^{2} + 2 \, B a^{2} d e + A a^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d^{2} + 2 \, A a^{2} d e\right )} x^{2} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.97 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {2}{7} \, B c^{2} d e x^{7} + \frac {1}{7} \, A c^{2} e^{2} x^{7} + \frac {1}{6} \, B c^{2} d^{2} x^{6} + \frac {1}{3} \, A c^{2} d e x^{6} + \frac {1}{3} \, B a c e^{2} x^{6} + \frac {1}{5} \, A c^{2} d^{2} x^{5} + \frac {4}{5} \, B a c d e x^{5} + \frac {2}{5} \, A a c e^{2} x^{5} + \frac {1}{2} \, B a c d^{2} x^{4} + A a c d e x^{4} + \frac {1}{4} \, B a^{2} e^{2} x^{4} + \frac {2}{3} \, A a c d^{2} x^{3} + \frac {2}{3} \, B a^{2} d e x^{3} + \frac {1}{3} \, A a^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{2} x^{2} + A a^{2} d e x^{2} + A a^{2} d^{2} x \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.82 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^2 \, dx=x^3\,\left (\frac {2\,B\,a^2\,d\,e}{3}+\frac {A\,a^2\,e^2}{3}+\frac {2\,A\,c\,a\,d^2}{3}\right )+x^6\,\left (\frac {B\,c^2\,d^2}{6}+\frac {A\,c^2\,d\,e}{3}+\frac {B\,a\,c\,e^2}{3}\right )+\frac {c\,x^5\,\left (A\,c\,d^2+4\,B\,a\,d\,e+2\,A\,a\,e^2\right )}{5}+\frac {a\,x^4\,\left (2\,B\,c\,d^2+4\,A\,c\,d\,e+B\,a\,e^2\right )}{4}+A\,a^2\,d^2\,x+\frac {a^2\,d\,x^2\,\left (2\,A\,e+B\,d\right )}{2}+\frac {c^2\,e\,x^7\,\left (A\,e+2\,B\,d\right )}{7}+\frac {B\,c^2\,e^2\,x^8}{8} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.98 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^2 \, dx=\frac {x \left (105 b \,c^{2} e^{2} x^{7}+120 a \,c^{2} e^{2} x^{6}+240 b \,c^{2} d e \,x^{6}+280 a b c \,e^{2} x^{5}+280 a \,c^{2} d e \,x^{5}+140 b \,c^{2} d^{2} x^{5}+336 a^{2} c \,e^{2} x^{4}+672 a b c d e \,x^{4}+168 a \,c^{2} d^{2} x^{4}+210 a^{2} b \,e^{2} x^{3}+840 a^{2} c d e \,x^{3}+420 a b c \,d^{2} x^{3}+280 a^{3} e^{2} x^{2}+560 a^{2} b d e \,x^{2}+560 a^{2} c \,d^{2} x^{2}+840 a^{3} d e x +420 a^{2} b \,d^{2} x +840 a^{3} d^{2}\right )}{840} \] Input:

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

Output:

(x*(840*a**3*d**2 + 840*a**3*d*e*x + 280*a**3*e**2*x**2 + 420*a**2*b*d**2* 
x + 560*a**2*b*d*e*x**2 + 210*a**2*b*e**2*x**3 + 560*a**2*c*d**2*x**2 + 84 
0*a**2*c*d*e*x**3 + 336*a**2*c*e**2*x**4 + 420*a*b*c*d**2*x**3 + 672*a*b*c 
*d*e*x**4 + 280*a*b*c*e**2*x**5 + 168*a*c**2*d**2*x**4 + 280*a*c**2*d*e*x* 
*5 + 120*a*c**2*e**2*x**6 + 140*b*c**2*d**2*x**5 + 240*b*c**2*d*e*x**6 + 1 
05*b*c**2*e**2*x**7))/840