3.2380 \(\int \frac{\sqrt{-1+x^{2/3}}}{\sqrt [3]{x}} \, dx\)

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 = 0.0120813, 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 \[ \left (x^{2/3}-1\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-1 + x^(2/3))^(3/2)

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Rubi in Sympy [A]  time = 1.63986, 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]  rubi_integrate((-1+x**(2/3))**(1/2)/x**(1/3),x)

[Out]

(x**(2/3) - 1)**(3/2)

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Mathematica [A]  time = 0.00608896, 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 = 0.008, size = 8, normalized size = 0.7 \[ \left ( -1+{x}^{{\frac{2}{3}}} \right ) ^{{\frac{3}{2}}} \]

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 = 1.49041, size = 9, normalized size = 0.82 \[{\left (x^{\frac{2}{3}} - 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^(2/3) - 1)/x^(1/3),x, algorithm="maxima")

[Out]

(x^(2/3) - 1)^(3/2)

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Fricas [A]  time = 0.916564, size = 9, normalized size = 0.82 \[{\left (x^{\frac{2}{3}} - 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^(2/3) - 1)/x^(1/3),x, algorithm="fricas")

[Out]

(x^(2/3) - 1)^(3/2)

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Sympy [A]  time = 0.539039, 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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GIAC/XCAS [A]  time = 0.218424, size = 9, normalized size = 0.82 \[{\left (x^{\frac{2}{3}} - 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^(2/3) - 1)/x^(1/3),x, algorithm="giac")

[Out]

(x^(2/3) - 1)^(3/2)