3.1315 \(\int (A+B x) (d+e x)^3 (a+c x^2)^3 \, dx\)

Optimal. Leaf size=334 \[ -\frac{c (d+e x)^7 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{7 e^8}+\frac{c^2 (d+e x)^9 \left (a B e^2-2 A c d e+7 B c d^2\right )}{3 e^8}-\frac{c^2 (d+e x)^8 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{8 e^8}-\frac{c (d+e x)^6 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8}+\frac{(d+e x)^5 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8}-\frac{(d+e x)^4 \left (a e^2+c d^2\right )^3 (B d-A e)}{4 e^8}-\frac{c^3 (d+e x)^{10} (7 B d-A e)}{10 e^8}+\frac{B c^3 (d+e x)^{11}}{11 e^8} \]

[Out]

-((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^4)/(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)

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Rubi [A]  time = 0.374761, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ -\frac{c (d+e x)^7 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{7 e^8}+\frac{c^2 (d+e x)^9 \left (a B e^2-2 A c d e+7 B c d^2\right )}{3 e^8}-\frac{c^2 (d+e x)^8 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{8 e^8}-\frac{c (d+e x)^6 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8}+\frac{(d+e x)^5 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8}-\frac{(d+e x)^4 \left (a e^2+c d^2\right )^3 (B d-A e)}{4 e^8}-\frac{c^3 (d+e x)^{10} (7 B d-A e)}{10 e^8}+\frac{B c^3 (d+e x)^{11}}{11 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^4)/(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)

Rule 772

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*} \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^3 \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3 (d+e x)^3}{e^7}+\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}{e^7}+\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}{e^7}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right ) (d+e x)^6}{e^7}+\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}{e^7}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^8}{e^7}+\frac{c^3 (-7 B d+A e) (d+e x)^9}{e^7}+\frac{B c^3 (d+e x)^{10}}{e^7}\right ) \, 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}\\ \end{align*}

Mathematica [A]  time = 0.0982875, size = 323, normalized size = 0.97 \[ \frac{1}{4} a^2 x^4 \left (a A e^3+3 a B d e^2+9 A c d^2 e+3 B c d^3\right )+a^2 d x^3 \left (a A e^2+a B d e+A c d^2\right )+\frac{1}{2} a^3 d^2 x^2 (3 A e+B d)+a^3 A d^3 x+\frac{1}{3} c^2 e x^9 \left (a B e^2+A c d e+B c d^2\right )+\frac{1}{8} c^2 x^8 \left (3 a A e^3+9 a B d e^2+3 A c d^2 e+B c d^3\right )+\frac{1}{7} c x^7 \left (A c d \left (9 a e^2+c d^2\right )+3 a B e \left (a e^2+3 c d^2\right )\right )+\frac{1}{2} a c x^6 \left (a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+\frac{1}{5} a x^5 \left (3 A c d \left (3 a e^2+c d^2\right )+a B e \left (a e^2+9 c d^2\right )\right )+\frac{1}{10} c^3 e^2 x^{10} (A e+3 B d)+\frac{1}{11} B c^3 e^3 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 353, normalized size = 1.1 \begin{align*}{\frac{B{c}^{3}{e}^{3}{x}^{11}}{11}}+{\frac{ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){c}^{3}{x}^{10}}{10}}+{\frac{ \left ( \left ( 3\,Ad{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\,Bd{e}^{2} \right ) a{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( A{c}^{3}{d}^{3}+3\, \left ( 3\,Ad{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\,Bd{e}^{2} \right ){a}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{d}^{3}a{c}^{2}+3\, \left ( 3\,Ad{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\,Bd{e}^{2} \right ){a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,A{d}^{3}{a}^{2}c+ \left ( 3\,Ad{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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^3*e^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

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Maxima [A]  time = 1.04086, size = 490, normalized size = 1.47 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.63206, size = 933, normalized size = 2.79 \begin{align*} \frac{1}{11} x^{11} e^{3} c^{3} B + \frac{3}{10} x^{10} e^{2} d c^{3} B + \frac{1}{10} x^{10} e^{3} c^{3} A + \frac{1}{3} x^{9} e d^{2} c^{3} B + \frac{1}{3} x^{9} e^{3} c^{2} a B + \frac{1}{3} x^{9} e^{2} d c^{3} A + \frac{1}{8} x^{8} d^{3} c^{3} B + \frac{9}{8} x^{8} e^{2} d c^{2} a B + \frac{3}{8} x^{8} e d^{2} c^{3} A + \frac{3}{8} x^{8} e^{3} c^{2} a A + \frac{9}{7} x^{7} e d^{2} c^{2} a B + \frac{3}{7} x^{7} e^{3} c a^{2} B + \frac{1}{7} x^{7} d^{3} c^{3} A + \frac{9}{7} x^{7} e^{2} d c^{2} a A + \frac{1}{2} x^{6} d^{3} c^{2} a B + \frac{3}{2} x^{6} e^{2} d c a^{2} B + \frac{3}{2} x^{6} e d^{2} c^{2} a A + \frac{1}{2} x^{6} e^{3} c a^{2} A + \frac{9}{5} x^{5} e d^{2} c a^{2} B + \frac{1}{5} x^{5} e^{3} a^{3} B + \frac{3}{5} x^{5} d^{3} c^{2} a A + \frac{9}{5} x^{5} e^{2} d c a^{2} A + \frac{3}{4} x^{4} d^{3} c a^{2} B + \frac{3}{4} x^{4} e^{2} d a^{3} B + \frac{9}{4} x^{4} e d^{2} c a^{2} A + \frac{1}{4} x^{4} e^{3} a^{3} A + x^{3} e d^{2} a^{3} B + x^{3} d^{3} c a^{2} A + x^{3} e^{2} d a^{3} A + \frac{1}{2} x^{2} d^{3} a^{3} B + \frac{3}{2} x^{2} e d^{2} a^{3} A + x d^{3} a^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.20832, size = 435, normalized size = 1.3 \begin{align*} 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} \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} \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} \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} \left (\frac{3 A a^{3} d^{2} e}{2} + \frac{B a^{3} d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Giac [A]  time = 1.17315, size = 544, normalized size = 1.63 \begin{align*} \frac{1}{11} \, B c^{3} x^{11} e^{3} + \frac{3}{10} \, B c^{3} d x^{10} e^{2} + \frac{1}{3} \, B c^{3} d^{2} x^{9} e + \frac{1}{8} \, B c^{3} d^{3} x^{8} + \frac{1}{10} \, A c^{3} x^{10} e^{3} + \frac{1}{3} \, A c^{3} d x^{9} e^{2} + \frac{3}{8} \, A c^{3} d^{2} x^{8} e + \frac{1}{7} \, A c^{3} d^{3} x^{7} + \frac{1}{3} \, B a c^{2} x^{9} e^{3} + \frac{9}{8} \, B a c^{2} d x^{8} e^{2} + \frac{9}{7} \, B a c^{2} d^{2} x^{7} e + \frac{1}{2} \, B a c^{2} d^{3} x^{6} + \frac{3}{8} \, A a c^{2} x^{8} e^{3} + \frac{9}{7} \, A a c^{2} d x^{7} e^{2} + \frac{3}{2} \, A a c^{2} d^{2} x^{6} e + \frac{3}{5} \, A a c^{2} d^{3} x^{5} + \frac{3}{7} \, B a^{2} c x^{7} e^{3} + \frac{3}{2} \, B a^{2} c d x^{6} e^{2} + \frac{9}{5} \, B a^{2} c d^{2} x^{5} e + \frac{3}{4} \, B a^{2} c d^{3} x^{4} + \frac{1}{2} \, A a^{2} c x^{6} e^{3} + \frac{9}{5} \, A a^{2} c d x^{5} e^{2} + \frac{9}{4} \, A a^{2} c d^{2} x^{4} e + A a^{2} c d^{3} x^{3} + \frac{1}{5} \, B a^{3} x^{5} e^{3} + \frac{3}{4} \, B a^{3} d x^{4} e^{2} + B a^{3} d^{2} x^{3} e + \frac{1}{2} \, B a^{3} d^{3} x^{2} + \frac{1}{4} \, A a^{3} x^{4} e^{3} + A a^{3} d x^{3} e^{2} + \frac{3}{2} \, A a^{3} d^{2} x^{2} e + A a^{3} d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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