3.2387 \(\int \frac{\sqrt{-1+x^{2/3}}}{\sqrt [3]{x}} \, dx\)

Optimal. Leaf size=11 \[ \left (x^{2/3}-1\right )^{3/2} \]

[Out]

(-1 + x^(2/3))^(3/2)

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Rubi [A]  time = 0.002669, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {261} \[ \left (x^{2/3}-1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x^(2/3)]/x^(1/3),x]

[Out]

(-1 + x^(2/3))^(3/2)

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 \frac{\sqrt{-1+x^{2/3}}}{\sqrt [3]{x}} \, dx &=\left (-1+x^{2/3}\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0031451, size = 11, normalized size = 1. \[ \left (x^{2/3}-1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + x^(2/3)]/x^(1/3),x]

[Out]

(-1 + x^(2/3))^(3/2)

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Maple [A]  time = 0.003, size = 8, normalized size = 0.7 \begin{align*} \left ( -1+{x}^{{\frac{2}{3}}} \right ) ^{{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1+x^(2/3))^(3/2)

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Maxima [A]  time = 0.987371, size = 9, normalized size = 0.82 \begin{align*}{\left (x^{\frac{2}{3}} - 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^(2/3) - 1)^(3/2)

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Fricas [A]  time = 6.30563, size = 28, normalized size = 2.55 \begin{align*}{\left (x^{\frac{2}{3}} - 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^(2/3) - 1)^(3/2)

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Sympy [B]  time = 0.201983, size = 24, normalized size = 2.18 \begin{align*} x^{\frac{2}{3}} \sqrt{x^{\frac{2}{3}} - 1} - \sqrt{x^{\frac{2}{3}} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(2/3)*sqrt(x**(2/3) - 1) - sqrt(x**(2/3) - 1)

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Giac [A]  time = 1.19553, size = 9, normalized size = 0.82 \begin{align*}{\left (x^{\frac{2}{3}} - 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^(2/3) - 1)^(3/2)