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 = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ \left (x^{2/3}-1\right )^{3/2} \]

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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fricas [A]  time = 1.91, size = 7, normalized size = 0.64 \[ {\left (x^{\frac {2}{3}} - 1\right )}^{\frac {3}{2}} \]

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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giac [A]  time = 0.18, size = 7, normalized size = 0.64 \[ {\left (x^{\frac {2}{3}} - 1\right )}^{\frac {3}{2}} \]

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)

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maple [A]  time = 0.01, size = 8, normalized size = 0.73 \[ \left (x^{\frac {2}{3}}-1\right )^{\frac {3}{2}} \]

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 = 0.52, size = 7, normalized size = 0.64 \[ {\left (x^{\frac {2}{3}} - 1\right )}^{\frac {3}{2}} \]

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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mupad [B]  time = 2.19, size = 7, normalized size = 0.64 \[ {\left (x^{2/3}-1\right )}^{3/2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(2/3) - 1)^(1/2)/x^(1/3),x)

[Out]

(x^(2/3) - 1)^(3/2)

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sympy [B]  time = 0.24, size = 24, normalized size = 2.18 \[ x^{\frac {2}{3}} \sqrt {x^{\frac {2}{3}} - 1} - \sqrt {x^{\frac {2}{3}} - 1} \]

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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