3.1297 \(\int (A+B x) (d+e x)^5 (a+c x^2)^2 \, dx\)

Optimal. Leaf size=206 \[ \frac{2 c (d+e x)^9 \left (a B e^2-2 A c d e+5 B c d^2\right )}{9 e^6}-\frac{c (d+e x)^8 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{4 e^6}+\frac{(d+e x)^7 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{7 e^6}-\frac{(d+e x)^6 \left (a e^2+c d^2\right )^2 (B d-A e)}{6 e^6}-\frac{c^2 (d+e x)^{10} (5 B d-A e)}{10 e^6}+\frac{B c^2 (d+e x)^{11}}{11 e^6} \]

[Out]

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

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Rubi [A]  time = 0.391214, antiderivative size = 206, 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} \[ \frac{2 c (d+e x)^9 \left (a B e^2-2 A c d e+5 B c d^2\right )}{9 e^6}-\frac{c (d+e x)^8 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{4 e^6}+\frac{(d+e x)^7 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{7 e^6}-\frac{(d+e x)^6 \left (a e^2+c d^2\right )^2 (B d-A e)}{6 e^6}-\frac{c^2 (d+e x)^{10} (5 B d-A e)}{10 e^6}+\frac{B c^2 (d+e x)^{11}}{11 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^5 \left (a+c x^2\right )^2 \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^2 (d+e x)^5}{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)^6}{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)^7}{e^5}-\frac{2 c \left (-5 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^8}{e^5}+\frac{c^2 (-5 B d+A e) (d+e x)^9}{e^5}+\frac{B c^2 (d+e x)^{10}}{e^5}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^6}{6 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)^7}{7 e^6}-\frac{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)^8}{4 e^6}+\frac{2 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^9}{9 e^6}-\frac{c^2 (5 B d-A e) (d+e x)^{10}}{10 e^6}+\frac{B c^2 (d+e x)^{11}}{11 e^6}\\ \end{align*}

Mathematica [A]  time = 0.106491, size = 390, normalized size = 1.89 \[ \frac{1}{7} e x^7 \left (a^2 B e^4+10 a A c d e^3+20 a B c d^2 e^2+10 A c^2 d^3 e+5 B c^2 d^4\right )+\frac{1}{6} x^6 \left (a^2 A e^5+5 a^2 B d e^4+20 a A c d^2 e^3+20 a B c d^3 e^2+5 A c^2 d^4 e+B c^2 d^5\right )+\frac{1}{5} d x^5 \left (5 a^2 A e^4+10 a^2 B d e^3+20 a A c d^2 e^2+10 a B c d^3 e+A c^2 d^4\right )+\frac{1}{2} a^2 d^4 x^2 (5 A e+B d)+a^2 A d^5 x+\frac{1}{9} c e^3 x^9 \left (2 a B e^2+5 A c d e+10 B c d^2\right )+\frac{1}{4} c e^2 x^8 \left (a A e^3+5 a B d e^2+5 A c d^2 e+5 B c d^3\right )+\frac{1}{2} a d^2 x^4 \left (5 a A e^3+5 a B d e^2+5 A c d^2 e+B c d^3\right )+\frac{1}{3} a d^3 x^3 \left (10 a A e^2+5 a B d e+2 A c d^2\right )+\frac{1}{10} c^2 e^4 x^{10} (A e+5 B d)+\frac{1}{11} B c^2 e^5 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*A*d^5*x + (a^2*d^4*(B*d + 5*A*e)*x^2)/2 + (a*d^3*(2*A*c*d^2 + 5*a*B*d*e + 10*a*A*e^2)*x^3)/3 + (a*d^2*(B*c
*d^3 + 5*A*c*d^2*e + 5*a*B*d*e^2 + 5*a*A*e^3)*x^4)/2 + (d*(A*c^2*d^4 + 10*a*B*c*d^3*e + 20*a*A*c*d^2*e^2 + 10*
a^2*B*d*e^3 + 5*a^2*A*e^4)*x^5)/5 + ((B*c^2*d^5 + 5*A*c^2*d^4*e + 20*a*B*c*d^3*e^2 + 20*a*A*c*d^2*e^3 + 5*a^2*
B*d*e^4 + a^2*A*e^5)*x^6)/6 + (e*(5*B*c^2*d^4 + 10*A*c^2*d^3*e + 20*a*B*c*d^2*e^2 + 10*a*A*c*d*e^3 + a^2*B*e^4
)*x^7)/7 + (c*e^2*(5*B*c*d^3 + 5*A*c*d^2*e + 5*a*B*d*e^2 + a*A*e^3)*x^8)/4 + (c*e^3*(10*B*c*d^2 + 5*A*c*d*e +
2*a*B*e^2)*x^9)/9 + (c^2*e^4*(5*B*d + A*e)*x^10)/10 + (B*c^2*e^5*x^11)/11

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Maple [B]  time = 0.003, size = 402, normalized size = 2. \begin{align*}{\frac{B{e}^{5}{c}^{2}{x}^{11}}{11}}+{\frac{ \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){c}^{2}{x}^{10}}{10}}+{\frac{ \left ( \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){c}^{2}+2\,B{e}^{5}ac \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){c}^{2}+2\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ) ac \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){c}^{2}+2\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ) ac+B{e}^{5}{a}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){c}^{2}+2\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ) ac+ \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{5}{c}^{2}+2\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ) ac+ \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ) ac+ \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,A{d}^{5}ac+ \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{2}{x}^{2}}{2}}+A{d}^{5}{a}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.997598, size = 554, normalized size = 2.69 \begin{align*} \frac{1}{11} \, B c^{2} e^{5} x^{11} + \frac{1}{10} \,{\left (5 \, B c^{2} d e^{4} + A c^{2} e^{5}\right )} x^{10} + \frac{1}{9} \,{\left (10 \, B c^{2} d^{2} e^{3} + 5 \, A c^{2} d e^{4} + 2 \, B a c e^{5}\right )} x^{9} + A a^{2} d^{5} x + \frac{1}{4} \,{\left (5 \, B c^{2} d^{3} e^{2} + 5 \, A c^{2} d^{2} e^{3} + 5 \, B a c d e^{4} + A a c e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (5 \, B c^{2} d^{4} e + 10 \, A c^{2} d^{3} e^{2} + 20 \, B a c d^{2} e^{3} + 10 \, A a c d e^{4} + B a^{2} e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (B c^{2} d^{5} + 5 \, A c^{2} d^{4} e + 20 \, B a c d^{3} e^{2} + 20 \, A a c d^{2} e^{3} + 5 \, B a^{2} d e^{4} + A a^{2} e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (A c^{2} d^{5} + 10 \, B a c d^{4} e + 20 \, A a c d^{3} e^{2} + 10 \, B a^{2} d^{2} e^{3} + 5 \, A a^{2} d e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (B a c d^{5} + 5 \, A a c d^{4} e + 5 \, B a^{2} d^{3} e^{2} + 5 \, A a^{2} d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (2 \, A a c d^{5} + 5 \, B a^{2} d^{4} e + 10 \, A a^{2} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d^{5} + 5 \, A a^{2} d^{4} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^2*e^5*x^11 + 1/10*(5*B*c^2*d*e^4 + A*c^2*e^5)*x^10 + 1/9*(10*B*c^2*d^2*e^3 + 5*A*c^2*d*e^4 + 2*B*a*c*
e^5)*x^9 + A*a^2*d^5*x + 1/4*(5*B*c^2*d^3*e^2 + 5*A*c^2*d^2*e^3 + 5*B*a*c*d*e^4 + A*a*c*e^5)*x^8 + 1/7*(5*B*c^
2*d^4*e + 10*A*c^2*d^3*e^2 + 20*B*a*c*d^2*e^3 + 10*A*a*c*d*e^4 + B*a^2*e^5)*x^7 + 1/6*(B*c^2*d^5 + 5*A*c^2*d^4
*e + 20*B*a*c*d^3*e^2 + 20*A*a*c*d^2*e^3 + 5*B*a^2*d*e^4 + A*a^2*e^5)*x^6 + 1/5*(A*c^2*d^5 + 10*B*a*c*d^4*e +
20*A*a*c*d^3*e^2 + 10*B*a^2*d^2*e^3 + 5*A*a^2*d*e^4)*x^5 + 1/2*(B*a*c*d^5 + 5*A*a*c*d^4*e + 5*B*a^2*d^3*e^2 +
5*A*a^2*d^2*e^3)*x^4 + 1/3*(2*A*a*c*d^5 + 5*B*a^2*d^4*e + 10*A*a^2*d^3*e^2)*x^3 + 1/2*(B*a^2*d^5 + 5*A*a^2*d^4
*e)*x^2

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Fricas [B]  time = 1.60704, size = 1062, normalized size = 5.16 \begin{align*} \frac{1}{11} x^{11} e^{5} c^{2} B + \frac{1}{2} x^{10} e^{4} d c^{2} B + \frac{1}{10} x^{10} e^{5} c^{2} A + \frac{10}{9} x^{9} e^{3} d^{2} c^{2} B + \frac{2}{9} x^{9} e^{5} c a B + \frac{5}{9} x^{9} e^{4} d c^{2} A + \frac{5}{4} x^{8} e^{2} d^{3} c^{2} B + \frac{5}{4} x^{8} e^{4} d c a B + \frac{5}{4} x^{8} e^{3} d^{2} c^{2} A + \frac{1}{4} x^{8} e^{5} c a A + \frac{5}{7} x^{7} e d^{4} c^{2} B + \frac{20}{7} x^{7} e^{3} d^{2} c a B + \frac{1}{7} x^{7} e^{5} a^{2} B + \frac{10}{7} x^{7} e^{2} d^{3} c^{2} A + \frac{10}{7} x^{7} e^{4} d c a A + \frac{1}{6} x^{6} d^{5} c^{2} B + \frac{10}{3} x^{6} e^{2} d^{3} c a B + \frac{5}{6} x^{6} e^{4} d a^{2} B + \frac{5}{6} x^{6} e d^{4} c^{2} A + \frac{10}{3} x^{6} e^{3} d^{2} c a A + \frac{1}{6} x^{6} e^{5} a^{2} A + 2 x^{5} e d^{4} c a B + 2 x^{5} e^{3} d^{2} a^{2} B + \frac{1}{5} x^{5} d^{5} c^{2} A + 4 x^{5} e^{2} d^{3} c a A + x^{5} e^{4} d a^{2} A + \frac{1}{2} x^{4} d^{5} c a B + \frac{5}{2} x^{4} e^{2} d^{3} a^{2} B + \frac{5}{2} x^{4} e d^{4} c a A + \frac{5}{2} x^{4} e^{3} d^{2} a^{2} A + \frac{5}{3} x^{3} e d^{4} a^{2} B + \frac{2}{3} x^{3} d^{5} c a A + \frac{10}{3} x^{3} e^{2} d^{3} a^{2} A + \frac{1}{2} x^{2} d^{5} a^{2} B + \frac{5}{2} x^{2} e d^{4} a^{2} A + x d^{5} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*e^5*c^2*B + 1/2*x^10*e^4*d*c^2*B + 1/10*x^10*e^5*c^2*A + 10/9*x^9*e^3*d^2*c^2*B + 2/9*x^9*e^5*c*a*B
+ 5/9*x^9*e^4*d*c^2*A + 5/4*x^8*e^2*d^3*c^2*B + 5/4*x^8*e^4*d*c*a*B + 5/4*x^8*e^3*d^2*c^2*A + 1/4*x^8*e^5*c*a*
A + 5/7*x^7*e*d^4*c^2*B + 20/7*x^7*e^3*d^2*c*a*B + 1/7*x^7*e^5*a^2*B + 10/7*x^7*e^2*d^3*c^2*A + 10/7*x^7*e^4*d
*c*a*A + 1/6*x^6*d^5*c^2*B + 10/3*x^6*e^2*d^3*c*a*B + 5/6*x^6*e^4*d*a^2*B + 5/6*x^6*e*d^4*c^2*A + 10/3*x^6*e^3
*d^2*c*a*A + 1/6*x^6*e^5*a^2*A + 2*x^5*e*d^4*c*a*B + 2*x^5*e^3*d^2*a^2*B + 1/5*x^5*d^5*c^2*A + 4*x^5*e^2*d^3*c
*a*A + x^5*e^4*d*a^2*A + 1/2*x^4*d^5*c*a*B + 5/2*x^4*e^2*d^3*a^2*B + 5/2*x^4*e*d^4*c*a*A + 5/2*x^4*e^3*d^2*a^2
*A + 5/3*x^3*e*d^4*a^2*B + 2/3*x^3*d^5*c*a*A + 10/3*x^3*e^2*d^3*a^2*A + 1/2*x^2*d^5*a^2*B + 5/2*x^2*e*d^4*a^2*
A + x*d^5*a^2*A

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Sympy [B]  time = 0.274473, size = 495, normalized size = 2.4 \begin{align*} A a^{2} d^{5} x + \frac{B c^{2} e^{5} x^{11}}{11} + x^{10} \left (\frac{A c^{2} e^{5}}{10} + \frac{B c^{2} d e^{4}}{2}\right ) + x^{9} \left (\frac{5 A c^{2} d e^{4}}{9} + \frac{2 B a c e^{5}}{9} + \frac{10 B c^{2} d^{2} e^{3}}{9}\right ) + x^{8} \left (\frac{A a c e^{5}}{4} + \frac{5 A c^{2} d^{2} e^{3}}{4} + \frac{5 B a c d e^{4}}{4} + \frac{5 B c^{2} d^{3} e^{2}}{4}\right ) + x^{7} \left (\frac{10 A a c d e^{4}}{7} + \frac{10 A c^{2} d^{3} e^{2}}{7} + \frac{B a^{2} e^{5}}{7} + \frac{20 B a c d^{2} e^{3}}{7} + \frac{5 B c^{2} d^{4} e}{7}\right ) + x^{6} \left (\frac{A a^{2} e^{5}}{6} + \frac{10 A a c d^{2} e^{3}}{3} + \frac{5 A c^{2} d^{4} e}{6} + \frac{5 B a^{2} d e^{4}}{6} + \frac{10 B a c d^{3} e^{2}}{3} + \frac{B c^{2} d^{5}}{6}\right ) + x^{5} \left (A a^{2} d e^{4} + 4 A a c d^{3} e^{2} + \frac{A c^{2} d^{5}}{5} + 2 B a^{2} d^{2} e^{3} + 2 B a c d^{4} e\right ) + x^{4} \left (\frac{5 A a^{2} d^{2} e^{3}}{2} + \frac{5 A a c d^{4} e}{2} + \frac{5 B a^{2} d^{3} e^{2}}{2} + \frac{B a c d^{5}}{2}\right ) + x^{3} \left (\frac{10 A a^{2} d^{3} e^{2}}{3} + \frac{2 A a c d^{5}}{3} + \frac{5 B a^{2} d^{4} e}{3}\right ) + x^{2} \left (\frac{5 A a^{2} d^{4} e}{2} + \frac{B a^{2} d^{5}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*d**5*x + B*c**2*e**5*x**11/11 + x**10*(A*c**2*e**5/10 + B*c**2*d*e**4/2) + x**9*(5*A*c**2*d*e**4/9 + 2*
B*a*c*e**5/9 + 10*B*c**2*d**2*e**3/9) + x**8*(A*a*c*e**5/4 + 5*A*c**2*d**2*e**3/4 + 5*B*a*c*d*e**4/4 + 5*B*c**
2*d**3*e**2/4) + x**7*(10*A*a*c*d*e**4/7 + 10*A*c**2*d**3*e**2/7 + B*a**2*e**5/7 + 20*B*a*c*d**2*e**3/7 + 5*B*
c**2*d**4*e/7) + x**6*(A*a**2*e**5/6 + 10*A*a*c*d**2*e**3/3 + 5*A*c**2*d**4*e/6 + 5*B*a**2*d*e**4/6 + 10*B*a*c
*d**3*e**2/3 + B*c**2*d**5/6) + x**5*(A*a**2*d*e**4 + 4*A*a*c*d**3*e**2 + A*c**2*d**5/5 + 2*B*a**2*d**2*e**3 +
2*B*a*c*d**4*e) + x**4*(5*A*a**2*d**2*e**3/2 + 5*A*a*c*d**4*e/2 + 5*B*a**2*d**3*e**2/2 + B*a*c*d**5/2) + x**3
*(10*A*a**2*d**3*e**2/3 + 2*A*a*c*d**5/3 + 5*B*a**2*d**4*e/3) + x**2*(5*A*a**2*d**4*e/2 + B*a**2*d**5/2)

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Giac [B]  time = 1.27503, size = 603, normalized size = 2.93 \begin{align*} \frac{1}{11} \, B c^{2} x^{11} e^{5} + \frac{1}{2} \, B c^{2} d x^{10} e^{4} + \frac{10}{9} \, B c^{2} d^{2} x^{9} e^{3} + \frac{5}{4} \, B c^{2} d^{3} x^{8} e^{2} + \frac{5}{7} \, B c^{2} d^{4} x^{7} e + \frac{1}{6} \, B c^{2} d^{5} x^{6} + \frac{1}{10} \, A c^{2} x^{10} e^{5} + \frac{5}{9} \, A c^{2} d x^{9} e^{4} + \frac{5}{4} \, A c^{2} d^{2} x^{8} e^{3} + \frac{10}{7} \, A c^{2} d^{3} x^{7} e^{2} + \frac{5}{6} \, A c^{2} d^{4} x^{6} e + \frac{1}{5} \, A c^{2} d^{5} x^{5} + \frac{2}{9} \, B a c x^{9} e^{5} + \frac{5}{4} \, B a c d x^{8} e^{4} + \frac{20}{7} \, B a c d^{2} x^{7} e^{3} + \frac{10}{3} \, B a c d^{3} x^{6} e^{2} + 2 \, B a c d^{4} x^{5} e + \frac{1}{2} \, B a c d^{5} x^{4} + \frac{1}{4} \, A a c x^{8} e^{5} + \frac{10}{7} \, A a c d x^{7} e^{4} + \frac{10}{3} \, A a c d^{2} x^{6} e^{3} + 4 \, A a c d^{3} x^{5} e^{2} + \frac{5}{2} \, A a c d^{4} x^{4} e + \frac{2}{3} \, A a c d^{5} x^{3} + \frac{1}{7} \, B a^{2} x^{7} e^{5} + \frac{5}{6} \, B a^{2} d x^{6} e^{4} + 2 \, B a^{2} d^{2} x^{5} e^{3} + \frac{5}{2} \, B a^{2} d^{3} x^{4} e^{2} + \frac{5}{3} \, B a^{2} d^{4} x^{3} e + \frac{1}{2} \, B a^{2} d^{5} x^{2} + \frac{1}{6} \, A a^{2} x^{6} e^{5} + A a^{2} d x^{5} e^{4} + \frac{5}{2} \, A a^{2} d^{2} x^{4} e^{3} + \frac{10}{3} \, A a^{2} d^{3} x^{3} e^{2} + \frac{5}{2} \, A a^{2} d^{4} x^{2} e + A a^{2} d^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^2*x^11*e^5 + 1/2*B*c^2*d*x^10*e^4 + 10/9*B*c^2*d^2*x^9*e^3 + 5/4*B*c^2*d^3*x^8*e^2 + 5/7*B*c^2*d^4*x^
7*e + 1/6*B*c^2*d^5*x^6 + 1/10*A*c^2*x^10*e^5 + 5/9*A*c^2*d*x^9*e^4 + 5/4*A*c^2*d^2*x^8*e^3 + 10/7*A*c^2*d^3*x
^7*e^2 + 5/6*A*c^2*d^4*x^6*e + 1/5*A*c^2*d^5*x^5 + 2/9*B*a*c*x^9*e^5 + 5/4*B*a*c*d*x^8*e^4 + 20/7*B*a*c*d^2*x^
7*e^3 + 10/3*B*a*c*d^3*x^6*e^2 + 2*B*a*c*d^4*x^5*e + 1/2*B*a*c*d^5*x^4 + 1/4*A*a*c*x^8*e^5 + 10/7*A*a*c*d*x^7*
e^4 + 10/3*A*a*c*d^2*x^6*e^3 + 4*A*a*c*d^3*x^5*e^2 + 5/2*A*a*c*d^4*x^4*e + 2/3*A*a*c*d^5*x^3 + 1/7*B*a^2*x^7*e
^5 + 5/6*B*a^2*d*x^6*e^4 + 2*B*a^2*d^2*x^5*e^3 + 5/2*B*a^2*d^3*x^4*e^2 + 5/3*B*a^2*d^4*x^3*e + 1/2*B*a^2*d^5*x
^2 + 1/6*A*a^2*x^6*e^5 + A*a^2*d*x^5*e^4 + 5/2*A*a^2*d^2*x^4*e^3 + 10/3*A*a^2*d^3*x^3*e^2 + 5/2*A*a^2*d^4*x^2*
e + A*a^2*d^5*x