3.277 \(\int x (a+b x^3)^5 \, dx\)

Optimal. Leaf size=66 \[ \frac{10}{11} a^2 b^3 x^{11}+\frac{5}{4} a^3 b^2 x^8+a^4 b x^5+\frac{a^5 x^2}{2}+\frac{5}{14} a b^4 x^{14}+\frac{b^5 x^{17}}{17} \]

[Out]

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

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Rubi [A]  time = 0.0234117, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {270} \[ \frac{10}{11} a^2 b^3 x^{11}+\frac{5}{4} a^3 b^2 x^8+a^4 b x^5+\frac{a^5 x^2}{2}+\frac{5}{14} a b^4 x^{14}+\frac{b^5 x^{17}}{17} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^3)^5,x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0019956, size = 66, normalized size = 1. \[ \frac{10}{11} a^2 b^3 x^{11}+\frac{5}{4} a^3 b^2 x^8+a^4 b x^5+\frac{a^5 x^2}{2}+\frac{5}{14} a b^4 x^{14}+\frac{b^5 x^{17}}{17} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^3)^5,x]

[Out]

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

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Maple [A]  time = 0., size = 57, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}{x}^{2}}{2}}+{a}^{4}b{x}^{5}+{\frac{5\,{a}^{3}{b}^{2}{x}^{8}}{4}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{11}}{11}}+{\frac{5\,a{b}^{4}{x}^{14}}{14}}+{\frac{{b}^{5}{x}^{17}}{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.975344, size = 76, normalized size = 1.15 \begin{align*} \frac{1}{17} \, b^{5} x^{17} + \frac{5}{14} \, a b^{4} x^{14} + \frac{10}{11} \, a^{2} b^{3} x^{11} + \frac{5}{4} \, a^{3} b^{2} x^{8} + a^{4} b x^{5} + \frac{1}{2} \, a^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.55903, size = 132, normalized size = 2. \begin{align*} \frac{1}{17} x^{17} b^{5} + \frac{5}{14} x^{14} b^{4} a + \frac{10}{11} x^{11} b^{3} a^{2} + \frac{5}{4} x^{8} b^{2} a^{3} + x^{5} b a^{4} + \frac{1}{2} x^{2} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.076707, size = 63, normalized size = 0.95 \begin{align*} \frac{a^{5} x^{2}}{2} + a^{4} b x^{5} + \frac{5 a^{3} b^{2} x^{8}}{4} + \frac{10 a^{2} b^{3} x^{11}}{11} + \frac{5 a b^{4} x^{14}}{14} + \frac{b^{5} x^{17}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*x**2/2 + a**4*b*x**5 + 5*a**3*b**2*x**8/4 + 10*a**2*b**3*x**11/11 + 5*a*b**4*x**14/14 + b**5*x**17/17

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Giac [A]  time = 1.09846, size = 76, normalized size = 1.15 \begin{align*} \frac{1}{17} \, b^{5} x^{17} + \frac{5}{14} \, a b^{4} x^{14} + \frac{10}{11} \, a^{2} b^{3} x^{11} + \frac{5}{4} \, a^{3} b^{2} x^{8} + a^{4} b x^{5} + \frac{1}{2} \, a^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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