3.221 \(\int x (a+b x^3)^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^2}{2}+\frac{2}{5} a b x^5+\frac{b^2 x^8}{8} \]

[Out]

(a^2*x^2)/2 + (2*a*b*x^5)/5 + (b^2*x^8)/8

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Rubi [A]  time = 0.0088576, antiderivative size = 30, 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} \[ \frac{a^2 x^2}{2}+\frac{2}{5} a b x^5+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^3)^2,x]

[Out]

(a^2*x^2)/2 + (2*a*b*x^5)/5 + (b^2*x^8)/8

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 )^2 \, dx &=\int \left (a^2 x+2 a b x^4+b^2 x^7\right ) \, dx\\ &=\frac{a^2 x^2}{2}+\frac{2}{5} a b x^5+\frac{b^2 x^8}{8}\\ \end{align*}

Mathematica [A]  time = 0.0013237, size = 30, normalized size = 1. \[ \frac{a^2 x^2}{2}+\frac{2}{5} a b x^5+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^3)^2,x]

[Out]

(a^2*x^2)/2 + (2*a*b*x^5)/5 + (b^2*x^8)/8

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Maple [A]  time = 0.002, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{2\,{x}^{5}ab}{5}}+{\frac{{b}^{2}{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^3+a)^2,x)

[Out]

1/2*a^2*x^2+2/5*x^5*a*b+1/8*b^2*x^8

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Maxima [A]  time = 0.945605, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{2}{5} \, a b x^{5} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/8*b^2*x^8 + 2/5*a*b*x^5 + 1/2*a^2*x^2

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Fricas [A]  time = 1.49516, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{8} x^{8} b^{2} + \frac{2}{5} x^{5} b a + \frac{1}{2} x^{2} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/8*x^8*b^2 + 2/5*x^5*b*a + 1/2*x^2*a^2

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Sympy [A]  time = 0.081009, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**3+a)**2,x)

[Out]

a**2*x**2/2 + 2*a*b*x**5/5 + b**2*x**8/8

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Giac [A]  time = 1.08669, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{2}{5} \, a b x^{5} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/8*b^2*x^8 + 2/5*a*b*x^5 + 1/2*a^2*x^2