3.462 \(\int x (1-x^3)^2 \, dx\)

Optimal. Leaf size=22 \[ \frac{x^8}{8}-\frac{2 x^5}{5}+\frac{x^2}{2} \]

[Out]

x^2/2 - (2*x^5)/5 + x^8/8

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Rubi [A]  time = 0.0053139, antiderivative size = 22, 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} \[ \frac{x^8}{8}-\frac{2 x^5}{5}+\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[x*(1 - x^3)^2,x]

[Out]

x^2/2 - (2*x^5)/5 + 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 (1-x^3\right )^2 \, dx &=\int \left (x-2 x^4+x^7\right ) \, dx\\ &=\frac{x^2}{2}-\frac{2 x^5}{5}+\frac{x^8}{8}\\ \end{align*}

Mathematica [A]  time = 0.0007037, size = 22, normalized size = 1. \[ \frac{x^8}{8}-\frac{2 x^5}{5}+\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(1 - x^3)^2,x]

[Out]

x^2/2 - (2*x^5)/5 + x^8/8

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Maple [A]  time = 0., size = 17, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}-{\frac{2\,{x}^{5}}{5}}+{\frac{{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-x^3+1)^2,x)

[Out]

1/2*x^2-2/5*x^5+1/8*x^8

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Maxima [A]  time = 1.1042, size = 22, normalized size = 1. \begin{align*} \frac{1}{8} \, x^{8} - \frac{2}{5} \, x^{5} + \frac{1}{2} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^3+1)^2,x, algorithm="maxima")

[Out]

1/8*x^8 - 2/5*x^5 + 1/2*x^2

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Fricas [A]  time = 1.03894, size = 39, normalized size = 1.77 \begin{align*} \frac{1}{8} x^{8} - \frac{2}{5} x^{5} + \frac{1}{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^3+1)^2,x, algorithm="fricas")

[Out]

1/8*x^8 - 2/5*x^5 + 1/2*x^2

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Sympy [A]  time = 0.051905, size = 15, normalized size = 0.68 \begin{align*} \frac{x^{8}}{8} - \frac{2 x^{5}}{5} + \frac{x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x**3+1)**2,x)

[Out]

x**8/8 - 2*x**5/5 + x**2/2

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Giac [A]  time = 1.10453, size = 22, normalized size = 1. \begin{align*} \frac{1}{8} \, x^{8} - \frac{2}{5} \, x^{5} + \frac{1}{2} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^3+1)^2,x, algorithm="giac")

[Out]

1/8*x^8 - 2/5*x^5 + 1/2*x^2