\(\int (A+B x) (d+e x)^4 (a+c x^2)^2 \, dx\) [1298]

Optimal result

Integrand size = 22, antiderivative size = 206 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^2 \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^5}{5 e^6}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^6}{6 e^6}-\frac {2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^7}{7 e^6}+\frac {c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^8}{4 e^6}-\frac {c^2 (5 B d-A e) (d+e x)^9}{9 e^6}+\frac {B c^2 (d+e x)^{10}}{10 e^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {786} \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^2 \, dx=\frac {c (d+e x)^8 \left (a B e^2-2 A c d e+5 B c d^2\right )}{4 e^6}+\frac {(d+e x)^6 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6}-\frac {(d+e x)^5 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6}-\frac {2 c (d+e x)^7 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}-\frac {c^2 (d+e x)^9 (5 B d-A e)}{9 e^6}+\frac {B c^2 (d+e x)^{10}}{10 e^6} \]

[In]

Int[(A + B*x)*(d + e*x)^4*(a + c*x^2)^2,x]

[Out]

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

Rule 786

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2 (d+e x)^4}{e^5}+\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)^5}{e^5}+\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)^6}{e^5}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^7}{e^5}+\frac {c^2 (-5 B d+A e) (d+e x)^8}{e^5}+\frac {B c^2 (d+e x)^9}{e^5}\right ) \, dx \\ & = -\frac {(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^5}{5 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)^6}{6 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)^7}{7 e^6}+\frac {c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^8}{4 e^6}-\frac {c^2 (5 B d-A e) (d+e x)^9}{9 e^6}+\frac {B c^2 (d+e x)^{10}}{10 e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.52 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^2 \, dx=a^2 A d^4 x+\frac {1}{2} a^2 d^3 (B d+4 A e) x^2+\frac {2}{3} a d^2 \left (A c d^2+2 a B d e+3 a A e^2\right ) x^3+\frac {1}{2} a d \left (B c d^3+4 A c d^2 e+3 a B d e^2+2 a A e^3\right ) x^4+\frac {1}{5} \left (A c^2 d^4+8 a B c d^3 e+12 a A c d^2 e^2+4 a^2 B d e^3+a^2 A e^4\right ) x^5+\frac {1}{6} \left (B c^2 d^4+4 A c^2 d^3 e+12 a B c d^2 e^2+8 a A c d e^3+a^2 B e^4\right ) x^6+\frac {2}{7} c e \left (2 B c d^3+3 A c d^2 e+4 a B d e^2+a A e^3\right ) x^7+\frac {1}{4} c e^2 \left (3 B c d^2+2 A c d e+a B e^2\right ) x^8+\frac {1}{9} c^2 e^3 (4 B d+A e) x^9+\frac {1}{10} B c^2 e^4 x^{10} \]

[In]

Integrate[(A + B*x)*(d + e*x)^4*(a + c*x^2)^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.59

method	result	size
default	\(\frac {B \,e^{4} c^{2} x^{10}}{10}+\frac {\left (e^{4} A +4 e^{3} d B \right ) c^{2} x^{9}}{9}+\frac {\left (\left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) c^{2}+2 B \,e^{4} a c \right ) x^{8}}{8}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c^{2}+2 \left (e^{4} A +4 e^{3} d B \right ) a c \right ) x^{7}}{7}+\frac {\left (\left (4 A \,d^{3} e +B \,d^{4}\right ) c^{2}+2 \left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) a c +B \,e^{4} a^{2}\right ) x^{6}}{6}+\frac {\left (d^{4} A \,c^{2}+2 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a c +\left (e^{4} A +4 e^{3} d B \right ) a^{2}\right ) x^{5}}{5}+\frac {\left (2 \left (4 A \,d^{3} e +B \,d^{4}\right ) a c +\left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) a^{2}\right ) x^{4}}{4}+\frac {\left (2 d^{4} A a c +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (4 A \,d^{3} e +B \,d^{4}\right ) a^{2} x^{2}}{2}+d^{4} A \,a^{2} x\)	\(327\)
norman	\(\frac {B \,e^{4} c^{2} x^{10}}{10}+\left (\frac {1}{9} A \,c^{2} e^{4}+\frac {4}{9} B \,c^{2} d \,e^{3}\right ) x^{9}+\left (\frac {1}{2} A \,c^{2} d \,e^{3}+\frac {1}{4} B \,e^{4} a c +\frac {3}{4} B \,c^{2} d^{2} e^{2}\right ) x^{8}+\left (\frac {2}{7} A a c \,e^{4}+\frac {6}{7} A \,c^{2} d^{2} e^{2}+\frac {8}{7} B a c d \,e^{3}+\frac {4}{7} B \,c^{2} d^{3} e \right ) x^{7}+\left (\frac {4}{3} A a c d \,e^{3}+\frac {2}{3} A \,c^{2} d^{3} e +\frac {1}{6} B \,e^{4} a^{2}+2 B a c \,d^{2} e^{2}+\frac {1}{6} B \,c^{2} d^{4}\right ) x^{6}+\left (\frac {1}{5} A \,a^{2} e^{4}+\frac {12}{5} A a c \,d^{2} e^{2}+\frac {1}{5} d^{4} A \,c^{2}+\frac {4}{5} B \,a^{2} d \,e^{3}+\frac {8}{5} B a c \,d^{3} e \right ) x^{5}+\left (A \,a^{2} d \,e^{3}+2 A a c \,d^{3} e +\frac {3}{2} B \,a^{2} d^{2} e^{2}+\frac {1}{2} B a c \,d^{4}\right ) x^{4}+\left (2 A \,a^{2} d^{2} e^{2}+\frac {2}{3} d^{4} A a c +\frac {4}{3} B \,a^{2} d^{3} e \right ) x^{3}+\left (2 A \,a^{2} d^{3} e +\frac {1}{2} B \,a^{2} d^{4}\right ) x^{2}+d^{4} A \,a^{2} x\)	\(334\)
gosper	\(\frac {1}{6} x^{6} B \,c^{2} d^{4}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{5} x^{5} d^{4} A \,c^{2}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+\frac {1}{10} B \,e^{4} c^{2} x^{10}+d^{4} A \,a^{2} x +\frac {1}{9} x^{9} A \,c^{2} e^{4}+\frac {12}{5} x^{5} A a c \,d^{2} e^{2}+\frac {8}{5} x^{5} B a c \,d^{3} e +2 x^{4} A a c \,d^{3} e +\frac {4}{3} x^{6} A a c d \,e^{3}+2 x^{6} B a c \,d^{2} e^{2}+\frac {8}{7} x^{7} B a c d \,e^{3}+\frac {4}{9} x^{9} B \,c^{2} d \,e^{3}+\frac {1}{2} x^{8} A \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} B \,e^{4} a c +\frac {3}{4} x^{8} B \,c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} A a c \,e^{4}+\frac {6}{7} x^{7} A \,c^{2} d^{2} e^{2}+\frac {4}{7} x^{7} B \,c^{2} d^{3} e +\frac {2}{3} x^{6} A \,c^{2} d^{3} e +\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+x^{4} A \,a^{2} d \,e^{3}+\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+\frac {1}{2} x^{4} B a c \,d^{4}+2 x^{3} A \,a^{2} d^{2} e^{2}+\frac {2}{3} x^{3} d^{4} A a c +\frac {4}{3} x^{3} B \,a^{2} d^{3} e +2 x^{2} A \,a^{2} d^{3} e\)	\(378\)
risch	\(\frac {1}{6} x^{6} B \,c^{2} d^{4}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{5} x^{5} d^{4} A \,c^{2}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+\frac {1}{10} B \,e^{4} c^{2} x^{10}+d^{4} A \,a^{2} x +\frac {1}{9} x^{9} A \,c^{2} e^{4}+\frac {12}{5} x^{5} A a c \,d^{2} e^{2}+\frac {8}{5} x^{5} B a c \,d^{3} e +2 x^{4} A a c \,d^{3} e +\frac {4}{3} x^{6} A a c d \,e^{3}+2 x^{6} B a c \,d^{2} e^{2}+\frac {8}{7} x^{7} B a c d \,e^{3}+\frac {4}{9} x^{9} B \,c^{2} d \,e^{3}+\frac {1}{2} x^{8} A \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} B \,e^{4} a c +\frac {3}{4} x^{8} B \,c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} A a c \,e^{4}+\frac {6}{7} x^{7} A \,c^{2} d^{2} e^{2}+\frac {4}{7} x^{7} B \,c^{2} d^{3} e +\frac {2}{3} x^{6} A \,c^{2} d^{3} e +\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+x^{4} A \,a^{2} d \,e^{3}+\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+\frac {1}{2} x^{4} B a c \,d^{4}+2 x^{3} A \,a^{2} d^{2} e^{2}+\frac {2}{3} x^{3} d^{4} A a c +\frac {4}{3} x^{3} B \,a^{2} d^{3} e +2 x^{2} A \,a^{2} d^{3} e\)	\(378\)
parallelrisch	\(\frac {1}{6} x^{6} B \,c^{2} d^{4}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{5} x^{5} d^{4} A \,c^{2}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+\frac {1}{10} B \,e^{4} c^{2} x^{10}+d^{4} A \,a^{2} x +\frac {1}{9} x^{9} A \,c^{2} e^{4}+\frac {12}{5} x^{5} A a c \,d^{2} e^{2}+\frac {8}{5} x^{5} B a c \,d^{3} e +2 x^{4} A a c \,d^{3} e +\frac {4}{3} x^{6} A a c d \,e^{3}+2 x^{6} B a c \,d^{2} e^{2}+\frac {8}{7} x^{7} B a c d \,e^{3}+\frac {4}{9} x^{9} B \,c^{2} d \,e^{3}+\frac {1}{2} x^{8} A \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} B \,e^{4} a c +\frac {3}{4} x^{8} B \,c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} A a c \,e^{4}+\frac {6}{7} x^{7} A \,c^{2} d^{2} e^{2}+\frac {4}{7} x^{7} B \,c^{2} d^{3} e +\frac {2}{3} x^{6} A \,c^{2} d^{3} e +\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+x^{4} A \,a^{2} d \,e^{3}+\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+\frac {1}{2} x^{4} B a c \,d^{4}+2 x^{3} A \,a^{2} d^{2} e^{2}+\frac {2}{3} x^{3} d^{4} A a c +\frac {4}{3} x^{3} B \,a^{2} d^{3} e +2 x^{2} A \,a^{2} d^{3} e\)	\(378\)

[In]

int((B*x+A)*(e*x+d)^4*(c*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.61 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^2 \, dx=\frac {1}{10} \, B c^{2} e^{4} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{2} d e^{3} + A c^{2} e^{4}\right )} x^{9} + \frac {1}{4} \, {\left (3 \, B c^{2} d^{2} e^{2} + 2 \, A c^{2} d e^{3} + B a c e^{4}\right )} x^{8} + A a^{2} d^{4} x + \frac {2}{7} \, {\left (2 \, B c^{2} d^{3} e + 3 \, A c^{2} d^{2} e^{2} + 4 \, B a c d e^{3} + A a c e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{4} + 4 \, A c^{2} d^{3} e + 12 \, B a c d^{2} e^{2} + 8 \, A a c d e^{3} + B a^{2} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A c^{2} d^{4} + 8 \, B a c d^{3} e + 12 \, A a c d^{2} e^{2} + 4 \, B a^{2} d e^{3} + A a^{2} e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (B a c d^{4} + 4 \, A a c d^{3} e + 3 \, B a^{2} d^{2} e^{2} + 2 \, A a^{2} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (A a c d^{4} + 2 \, B a^{2} d^{3} e + 3 \, A a^{2} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d^{4} + 4 \, A a^{2} d^{3} e\right )} x^{2} \]

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+a)^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.93 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^2 \, dx=A a^{2} d^{4} x + \frac {B c^{2} e^{4} x^{10}}{10} + x^{9} \left (\frac {A c^{2} e^{4}}{9} + \frac {4 B c^{2} d e^{3}}{9}\right ) + x^{8} \left (\frac {A c^{2} d e^{3}}{2} + \frac {B a c e^{4}}{4} + \frac {3 B c^{2} d^{2} e^{2}}{4}\right ) + x^{7} \cdot \left (\frac {2 A a c e^{4}}{7} + \frac {6 A c^{2} d^{2} e^{2}}{7} + \frac {8 B a c d e^{3}}{7} + \frac {4 B c^{2} d^{3} e}{7}\right ) + x^{6} \cdot \left (\frac {4 A a c d e^{3}}{3} + \frac {2 A c^{2} d^{3} e}{3} + \frac {B a^{2} e^{4}}{6} + 2 B a c d^{2} e^{2} + \frac {B c^{2} d^{4}}{6}\right ) + x^{5} \left (\frac {A a^{2} e^{4}}{5} + \frac {12 A a c d^{2} e^{2}}{5} + \frac {A c^{2} d^{4}}{5} + \frac {4 B a^{2} d e^{3}}{5} + \frac {8 B a c d^{3} e}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 2 A a c d^{3} e + \frac {3 B a^{2} d^{2} e^{2}}{2} + \frac {B a c d^{4}}{2}\right ) + x^{3} \cdot \left (2 A a^{2} d^{2} e^{2} + \frac {2 A a c d^{4}}{3} + \frac {4 B a^{2} d^{3} e}{3}\right ) + x^{2} \cdot \left (2 A a^{2} d^{3} e + \frac {B a^{2} d^{4}}{2}\right ) \]

[In]

integrate((B*x+A)*(e*x+d)**4*(c*x**2+a)**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.61 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^2 \, dx=\frac {1}{10} \, B c^{2} e^{4} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{2} d e^{3} + A c^{2} e^{4}\right )} x^{9} + \frac {1}{4} \, {\left (3 \, B c^{2} d^{2} e^{2} + 2 \, A c^{2} d e^{3} + B a c e^{4}\right )} x^{8} + A a^{2} d^{4} x + \frac {2}{7} \, {\left (2 \, B c^{2} d^{3} e + 3 \, A c^{2} d^{2} e^{2} + 4 \, B a c d e^{3} + A a c e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{4} + 4 \, A c^{2} d^{3} e + 12 \, B a c d^{2} e^{2} + 8 \, A a c d e^{3} + B a^{2} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A c^{2} d^{4} + 8 \, B a c d^{3} e + 12 \, A a c d^{2} e^{2} + 4 \, B a^{2} d e^{3} + A a^{2} e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (B a c d^{4} + 4 \, A a c d^{3} e + 3 \, B a^{2} d^{2} e^{2} + 2 \, A a^{2} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (A a c d^{4} + 2 \, B a^{2} d^{3} e + 3 \, A a^{2} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d^{4} + 4 \, A a^{2} d^{3} e\right )} x^{2} \]

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+a)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.83 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^2 \, dx=\frac {1}{10} \, B c^{2} e^{4} x^{10} + \frac {4}{9} \, B c^{2} d e^{3} x^{9} + \frac {1}{9} \, A c^{2} e^{4} x^{9} + \frac {3}{4} \, B c^{2} d^{2} e^{2} x^{8} + \frac {1}{2} \, A c^{2} d e^{3} x^{8} + \frac {1}{4} \, B a c e^{4} x^{8} + \frac {4}{7} \, B c^{2} d^{3} e x^{7} + \frac {6}{7} \, A c^{2} d^{2} e^{2} x^{7} + \frac {8}{7} \, B a c d e^{3} x^{7} + \frac {2}{7} \, A a c e^{4} x^{7} + \frac {1}{6} \, B c^{2} d^{4} x^{6} + \frac {2}{3} \, A c^{2} d^{3} e x^{6} + 2 \, B a c d^{2} e^{2} x^{6} + \frac {4}{3} \, A a c d e^{3} x^{6} + \frac {1}{6} \, B a^{2} e^{4} x^{6} + \frac {1}{5} \, A c^{2} d^{4} x^{5} + \frac {8}{5} \, B a c d^{3} e x^{5} + \frac {12}{5} \, A a c d^{2} e^{2} x^{5} + \frac {4}{5} \, B a^{2} d e^{3} x^{5} + \frac {1}{5} \, A a^{2} e^{4} x^{5} + \frac {1}{2} \, B a c d^{4} x^{4} + 2 \, A a c d^{3} e x^{4} + \frac {3}{2} \, B a^{2} d^{2} e^{2} x^{4} + A a^{2} d e^{3} x^{4} + \frac {2}{3} \, A a c d^{4} x^{3} + \frac {4}{3} \, B a^{2} d^{3} e x^{3} + 2 \, A a^{2} d^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{4} x^{2} + 2 \, A a^{2} d^{3} e x^{2} + A a^{2} d^{4} x \]

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+a)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.45 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^2 \, dx=x^5\,\left (\frac {4\,B\,a^2\,d\,e^3}{5}+\frac {A\,a^2\,e^4}{5}+\frac {8\,B\,a\,c\,d^3\,e}{5}+\frac {12\,A\,a\,c\,d^2\,e^2}{5}+\frac {A\,c^2\,d^4}{5}\right )+x^6\,\left (\frac {B\,a^2\,e^4}{6}+2\,B\,a\,c\,d^2\,e^2+\frac {4\,A\,a\,c\,d\,e^3}{3}+\frac {B\,c^2\,d^4}{6}+\frac {2\,A\,c^2\,d^3\,e}{3}\right )+\frac {2\,a\,d^2\,x^3\,\left (A\,c\,d^2+2\,B\,a\,d\,e+3\,A\,a\,e^2\right )}{3}+\frac {c\,e^2\,x^8\,\left (3\,B\,c\,d^2+2\,A\,c\,d\,e+B\,a\,e^2\right )}{4}+\frac {a^2\,d^3\,x^2\,\left (4\,A\,e+B\,d\right )}{2}+\frac {c^2\,e^3\,x^9\,\left (A\,e+4\,B\,d\right )}{9}+\frac {a\,d\,x^4\,\left (B\,c\,d^3+4\,A\,c\,d^2\,e+3\,B\,a\,d\,e^2+2\,A\,a\,e^3\right )}{2}+\frac {2\,c\,e\,x^7\,\left (2\,B\,c\,d^3+3\,A\,c\,d^2\,e+4\,B\,a\,d\,e^2+A\,a\,e^3\right )}{7}+A\,a^2\,d^4\,x+\frac {B\,c^2\,e^4\,x^{10}}{10} \]

[In]

int((a + c*x^2)^2*(A + B*x)*(d + e*x)^4,x)

[Out]

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