3.701 \(\int x (-1+x^2)^{7/3} \, dx\)

Optimal. Leaf size=13 \[ \frac{3}{20} \left (x^2-1\right )^{10/3} \]

[Out]

(3*(-1 + x^2)^(10/3))/20

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Rubi [A]  time = 0.0021335, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {261} \[ \frac{3}{20} \left (x^2-1\right )^{10/3} \]

Antiderivative was successfully verified.

[In]

Int[x*(-1 + x^2)^(7/3),x]

[Out]

(3*(-1 + x^2)^(10/3))/20

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x \left (-1+x^2\right )^{7/3} \, dx &=\frac{3}{20} \left (-1+x^2\right )^{10/3}\\ \end{align*}

Mathematica [A]  time = 0.0044398, size = 13, normalized size = 1. \[ \frac{3}{20} \left (x^2-1\right )^{10/3} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(-1 + x^2)^(7/3),x]

[Out]

(3*(-1 + x^2)^(10/3))/20

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Maple [A]  time = 0.001, size = 16, normalized size = 1.2 \begin{align*}{\frac{ \left ( 3+3\,x \right ) \left ( -1+x \right ) }{20} \left ({x}^{2}-1 \right ) ^{{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2-1)^(7/3),x)

[Out]

3/20*(1+x)*(-1+x)*(x^2-1)^(7/3)

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Maxima [A]  time = 1.84928, size = 12, normalized size = 0.92 \begin{align*} \frac{3}{20} \,{\left (x^{2} - 1\right )}^{\frac{10}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^(7/3),x, algorithm="maxima")

[Out]

3/20*(x^2 - 1)^(10/3)

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Fricas [B]  time = 1.65653, size = 65, normalized size = 5. \begin{align*} \frac{3}{20} \,{\left (x^{6} - 3 \, x^{4} + 3 \, x^{2} - 1\right )}{\left (x^{2} - 1\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^(7/3),x, algorithm="fricas")

[Out]

3/20*(x^6 - 3*x^4 + 3*x^2 - 1)*(x^2 - 1)^(1/3)

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Sympy [B]  time = 2.75557, size = 56, normalized size = 4.31 \begin{align*} \frac{3 x^{6} \sqrt [3]{x^{2} - 1}}{20} - \frac{9 x^{4} \sqrt [3]{x^{2} - 1}}{20} + \frac{9 x^{2} \sqrt [3]{x^{2} - 1}}{20} - \frac{3 \sqrt [3]{x^{2} - 1}}{20} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x**2-1)**(7/3),x)

[Out]

3*x**6*(x**2 - 1)**(1/3)/20 - 9*x**4*(x**2 - 1)**(1/3)/20 + 9*x**2*(x**2 - 1)**(1/3)/20 - 3*(x**2 - 1)**(1/3)/
20

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Giac [A]  time = 2.7084, size = 12, normalized size = 0.92 \begin{align*} \frac{3}{20} \,{\left (x^{2} - 1\right )}^{\frac{10}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^(7/3),x, algorithm="giac")

[Out]

3/20*(x^2 - 1)^(10/3)