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

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

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Rubi [A]  time = 0.39, antiderivative size = 206, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.10, size = 390, normalized size = 1.89 \begin {gather*} \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}{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}{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}{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}{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}{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}{10} c^2 e^4 x^{10} (A e+5 B d)+\frac {1}{11} B c^2 e^5 x^{11} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^5 \left (a+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.36, size = 465, normalized size = 2.26 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.19, size = 447, normalized size = 2.17 \begin {gather*} \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 {gather*}

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

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maple [B]  time = 0.04, size = 402, normalized size = 1.95 \begin {gather*} \frac {B \,c^{2} e^{5} x^{11}}{11}+\frac {\left (A \,e^{5}+5 B d \,e^{4}\right ) c^{2} x^{10}}{10}+A \,a^{2} d^{5} x +\frac {\left (2 B a c \,e^{5}+\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) c^{2}\right ) x^{9}}{9}+\frac {\left (2 \left (A \,e^{5}+5 B d \,e^{4}\right ) a c +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) c^{2}\right ) x^{8}}{8}+\frac {\left (B \,a^{2} e^{5}+2 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a c +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) c^{2}\right ) x^{7}}{7}+\frac {\left (\left (A \,e^{5}+5 B d \,e^{4}\right ) a^{2}+2 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a c +\left (5 A \,d^{4} e +B \,d^{5}\right ) c^{2}\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{5}+\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{2}+2 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a c \right ) x^{5}}{5}+\frac {\left (5 A \,d^{4} e +B \,d^{5}\right ) a^{2} x^{2}}{2}+\frac {\left (\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{2}+2 \left (5 A \,d^{4} e +B \,d^{5}\right ) a c \right ) x^{4}}{4}+\frac {\left (2 A a c \,d^{5}+\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a^{2}\right ) x^{3}}{3} \end {gather*}

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 = 0.53, size = 410, normalized size = 1.99 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 1.86, size = 374, normalized size = 1.82 \begin {gather*} x^5\,\left (2\,B\,a^2\,d^2\,e^3+A\,a^2\,d\,e^4+2\,B\,a\,c\,d^4\,e+4\,A\,a\,c\,d^3\,e^2+\frac {A\,c^2\,d^5}{5}\right )+x^7\,\left (\frac {B\,a^2\,e^5}{7}+\frac {20\,B\,a\,c\,d^2\,e^3}{7}+\frac {10\,A\,a\,c\,d\,e^4}{7}+\frac {5\,B\,c^2\,d^4\,e}{7}+\frac {10\,A\,c^2\,d^3\,e^2}{7}\right )+x^6\,\left (\frac {5\,B\,a^2\,d\,e^4}{6}+\frac {A\,a^2\,e^5}{6}+\frac {10\,B\,a\,c\,d^3\,e^2}{3}+\frac {10\,A\,a\,c\,d^2\,e^3}{3}+\frac {B\,c^2\,d^5}{6}+\frac {5\,A\,c^2\,d^4\,e}{6}\right )+\frac {a\,d^3\,x^3\,\left (2\,A\,c\,d^2+5\,B\,a\,d\,e+10\,A\,a\,e^2\right )}{3}+\frac {c\,e^3\,x^9\,\left (10\,B\,c\,d^2+5\,A\,c\,d\,e+2\,B\,a\,e^2\right )}{9}+\frac {a^2\,d^4\,x^2\,\left (5\,A\,e+B\,d\right )}{2}+\frac {c^2\,e^4\,x^{10}\,\left (A\,e+5\,B\,d\right )}{10}+A\,a^2\,d^5\,x+\frac {a\,d^2\,x^4\,\left (B\,c\,d^3+5\,A\,c\,d^2\,e+5\,B\,a\,d\,e^2+5\,A\,a\,e^3\right )}{2}+\frac {c\,e^2\,x^8\,\left (5\,B\,c\,d^3+5\,A\,c\,d^2\,e+5\,B\,a\,d\,e^2+A\,a\,e^3\right )}{4}+\frac {B\,c^2\,e^5\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.14, size = 495, normalized size = 2.40 \begin {gather*} 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 {gather*}

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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