3.31 \(\int x (a+b x^3)^5 (A+B x^3) \, dx\)

Optimal. Leaf size=117 \[ \frac{10}{11} a^2 b^2 x^{11} (a B+A b)+\frac{5}{8} a^3 b x^8 (a B+2 A b)+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{1}{2} a^5 A x^2+\frac{1}{17} b^4 x^{17} (5 a B+A b)+\frac{5}{14} a b^3 x^{14} (2 a B+A b)+\frac{1}{20} b^5 B x^{20} \]

[Out]

(a^5*A*x^2)/2 + (a^4*(5*A*b + a*B)*x^5)/5 + (5*a^3*b*(2*A*b + a*B)*x^8)/8 + (10*a^2*b^2*(A*b + a*B)*x^11)/11 +
(5*a*b^3*(A*b + 2*a*B)*x^14)/14 + (b^4*(A*b + 5*a*B)*x^17)/17 + (b^5*B*x^20)/20

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Rubi [A]  time = 0.0632876, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {448} \[ \frac{10}{11} a^2 b^2 x^{11} (a B+A b)+\frac{5}{8} a^3 b x^8 (a B+2 A b)+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{1}{2} a^5 A x^2+\frac{1}{17} b^4 x^{17} (5 a B+A b)+\frac{5}{14} a b^3 x^{14} (2 a B+A b)+\frac{1}{20} b^5 B x^{20} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(a^5*A*x^2)/2 + (a^4*(5*A*b + a*B)*x^5)/5 + (5*a^3*b*(2*A*b + a*B)*x^8)/8 + (10*a^2*b^2*(A*b + a*B)*x^11)/11 +
(5*a*b^3*(A*b + 2*a*B)*x^14)/14 + (b^4*(A*b + 5*a*B)*x^17)/17 + (b^5*B*x^20)/20

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int x \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx &=\int \left (a^5 A x+a^4 (5 A b+a B) x^4+5 a^3 b (2 A b+a B) x^7+10 a^2 b^2 (A b+a B) x^{10}+5 a b^3 (A b+2 a B) x^{13}+b^4 (A b+5 a B) x^{16}+b^5 B x^{19}\right ) \, dx\\ &=\frac{1}{2} a^5 A x^2+\frac{1}{5} a^4 (5 A b+a B) x^5+\frac{5}{8} a^3 b (2 A b+a B) x^8+\frac{10}{11} a^2 b^2 (A b+a B) x^{11}+\frac{5}{14} a b^3 (A b+2 a B) x^{14}+\frac{1}{17} b^4 (A b+5 a B) x^{17}+\frac{1}{20} b^5 B x^{20}\\ \end{align*}

Mathematica [A]  time = 0.0156807, size = 117, normalized size = 1. \[ \frac{10}{11} a^2 b^2 x^{11} (a B+A b)+\frac{5}{8} a^3 b x^8 (a B+2 A b)+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{1}{2} a^5 A x^2+\frac{1}{17} b^4 x^{17} (5 a B+A b)+\frac{5}{14} a b^3 x^{14} (2 a B+A b)+\frac{1}{20} b^5 B x^{20} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(a^5*A*x^2)/2 + (a^4*(5*A*b + a*B)*x^5)/5 + (5*a^3*b*(2*A*b + a*B)*x^8)/8 + (10*a^2*b^2*(A*b + a*B)*x^11)/11 +
(5*a*b^3*(A*b + 2*a*B)*x^14)/14 + (b^4*(A*b + 5*a*B)*x^17)/17 + (b^5*B*x^20)/20

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Maple [A]  time = 0.001, size = 124, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{20}}{20}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{17}}{17}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{14}}{14}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{8}}{8}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{5}}{5}}+{\frac{{a}^{5}A{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^3+a)^5*(B*x^3+A),x)

[Out]

1/20*b^5*B*x^20+1/17*(A*b^5+5*B*a*b^4)*x^17+1/14*(5*A*a*b^4+10*B*a^2*b^3)*x^14+1/11*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^11+1/8*(10*A*a^3*b^2+5*B*a^4*b)*x^8+1/5*(5*A*a^4*b+B*a^5)*x^5+1/2*a^5*A*x^2

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Maxima [A]  time = 1.39543, size = 161, normalized size = 1.38 \begin{align*} \frac{1}{20} \, B b^{5} x^{20} + \frac{1}{17} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{17} + \frac{5}{14} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{14} + \frac{10}{11} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{11} + \frac{5}{8} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac{1}{2} \, A a^{5} x^{2} + \frac{1}{5} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a)^5*(B*x^3+A),x, algorithm="maxima")

[Out]

1/20*B*b^5*x^20 + 1/17*(5*B*a*b^4 + A*b^5)*x^17 + 5/14*(2*B*a^2*b^3 + A*a*b^4)*x^14 + 10/11*(B*a^3*b^2 + A*a^2
*b^3)*x^11 + 5/8*(B*a^4*b + 2*A*a^3*b^2)*x^8 + 1/2*A*a^5*x^2 + 1/5*(B*a^5 + 5*A*a^4*b)*x^5

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Fricas [A]  time = 1.22414, size = 305, normalized size = 2.61 \begin{align*} \frac{1}{20} x^{20} b^{5} B + \frac{5}{17} x^{17} b^{4} a B + \frac{1}{17} x^{17} b^{5} A + \frac{5}{7} x^{14} b^{3} a^{2} B + \frac{5}{14} x^{14} b^{4} a A + \frac{10}{11} x^{11} b^{2} a^{3} B + \frac{10}{11} x^{11} b^{3} a^{2} A + \frac{5}{8} x^{8} b a^{4} B + \frac{5}{4} x^{8} b^{2} a^{3} A + \frac{1}{5} x^{5} a^{5} B + x^{5} b a^{4} A + \frac{1}{2} x^{2} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a)^5*(B*x^3+A),x, algorithm="fricas")

[Out]

1/20*x^20*b^5*B + 5/17*x^17*b^4*a*B + 1/17*x^17*b^5*A + 5/7*x^14*b^3*a^2*B + 5/14*x^14*b^4*a*A + 10/11*x^11*b^
2*a^3*B + 10/11*x^11*b^3*a^2*A + 5/8*x^8*b*a^4*B + 5/4*x^8*b^2*a^3*A + 1/5*x^5*a^5*B + x^5*b*a^4*A + 1/2*x^2*a
^5*A

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Sympy [A]  time = 0.086919, size = 134, normalized size = 1.15 \begin{align*} \frac{A a^{5} x^{2}}{2} + \frac{B b^{5} x^{20}}{20} + x^{17} \left (\frac{A b^{5}}{17} + \frac{5 B a b^{4}}{17}\right ) + x^{14} \left (\frac{5 A a b^{4}}{14} + \frac{5 B a^{2} b^{3}}{7}\right ) + x^{11} \left (\frac{10 A a^{2} b^{3}}{11} + \frac{10 B a^{3} b^{2}}{11}\right ) + x^{8} \left (\frac{5 A a^{3} b^{2}}{4} + \frac{5 B a^{4} b}{8}\right ) + x^{5} \left (A a^{4} b + \frac{B a^{5}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**3+a)**5*(B*x**3+A),x)

[Out]

A*a**5*x**2/2 + B*b**5*x**20/20 + x**17*(A*b**5/17 + 5*B*a*b**4/17) + x**14*(5*A*a*b**4/14 + 5*B*a**2*b**3/7)
+ x**11*(10*A*a**2*b**3/11 + 10*B*a**3*b**2/11) + x**8*(5*A*a**3*b**2/4 + 5*B*a**4*b/8) + x**5*(A*a**4*b + B*a
**5/5)

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Giac [A]  time = 1.1619, size = 167, normalized size = 1.43 \begin{align*} \frac{1}{20} \, B b^{5} x^{20} + \frac{5}{17} \, B a b^{4} x^{17} + \frac{1}{17} \, A b^{5} x^{17} + \frac{5}{7} \, B a^{2} b^{3} x^{14} + \frac{5}{14} \, A a b^{4} x^{14} + \frac{10}{11} \, B a^{3} b^{2} x^{11} + \frac{10}{11} \, A a^{2} b^{3} x^{11} + \frac{5}{8} \, B a^{4} b x^{8} + \frac{5}{4} \, A a^{3} b^{2} x^{8} + \frac{1}{5} \, B a^{5} x^{5} + A a^{4} b x^{5} + \frac{1}{2} \, A a^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a)^5*(B*x^3+A),x, algorithm="giac")

[Out]

1/20*B*b^5*x^20 + 5/17*B*a*b^4*x^17 + 1/17*A*b^5*x^17 + 5/7*B*a^2*b^3*x^14 + 5/14*A*a*b^4*x^14 + 10/11*B*a^3*b
^2*x^11 + 10/11*A*a^2*b^3*x^11 + 5/8*B*a^4*b*x^8 + 5/4*A*a^3*b^2*x^8 + 1/5*B*a^5*x^5 + A*a^4*b*x^5 + 1/2*A*a^5
*x^2