3.2376 \(\int \frac{x^{2/3}}{1+\sqrt [3]{x}} \, dx\)

Optimal. Leaf size=39 \[ \frac{3 x^{4/3}}{4}+\frac{3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]

[Out]

-3*x^(1/3) + (3*x^(2/3))/2 - x + (3*x^(4/3))/4 + 3*Log[1 + x^(1/3)]

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Rubi [A]  time = 0.0427283, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{3 x^{4/3}}{4}+\frac{3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^(2/3)/(1 + x^(1/3)),x]

[Out]

-3*x^(1/3) + (3*x^(2/3))/2 - x + (3*x^(4/3))/4 + 3*Log[1 + x^(1/3)]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 x^{\frac{4}{3}}}{4} - 3 \sqrt [3]{x} - x + 3 \log{\left (\sqrt [3]{x} + 1 \right )} + 3 \int ^{\sqrt [3]{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(2/3)/(1+x**(1/3)),x)

[Out]

3*x**(4/3)/4 - 3*x**(1/3) - x + 3*log(x**(1/3) + 1) + 3*Integral(x, (x, x**(1/3)
))

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Mathematica [A]  time = 0.0106657, size = 39, normalized size = 1. \[ \frac{3 x^{4/3}}{4}+\frac{3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-3*x^(1/3) + (3*x^(2/3))/2 - x + (3*x^(4/3))/4 + 3*Log[1 + x^(1/3)]

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Maple [A]  time = 0.003, size = 28, normalized size = 0.7 \[ -3\,\sqrt [3]{x}+{\frac{3}{2}{x}^{{\frac{2}{3}}}}-x+{\frac{3}{4}{x}^{{\frac{4}{3}}}}+3\,\ln \left ( 1+\sqrt [3]{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(2/3)/(1+x^(1/3)),x)

[Out]

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

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Maxima [A]  time = 1.43089, size = 57, normalized size = 1.46 \[ \frac{3}{4} \,{\left (x^{\frac{1}{3}} + 1\right )}^{4} - 4 \,{\left (x^{\frac{1}{3}} + 1\right )}^{3} + 9 \,{\left (x^{\frac{1}{3}} + 1\right )}^{2} - 12 \, x^{\frac{1}{3}} + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) - 12 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*(x^(1/3) + 1)^4 - 4*(x^(1/3) + 1)^3 + 9*(x^(1/3) + 1)^2 - 12*x^(1/3) + 3*log
(x^(1/3) + 1) - 12

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Fricas [A]  time = 0.221714, size = 34, normalized size = 0.87 \[ \frac{3}{4} \,{\left (x - 4\right )} x^{\frac{1}{3}} - x + \frac{3}{2} \, x^{\frac{2}{3}} + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.597698, size = 34, normalized size = 0.87 \[ \frac{3 x^{\frac{4}{3}}}{4} + \frac{3 x^{\frac{2}{3}}}{2} - 3 \sqrt [3]{x} - x + 3 \log{\left (\sqrt [3]{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(2/3)/(1+x**(1/3)),x)

[Out]

3*x**(4/3)/4 + 3*x**(2/3)/2 - 3*x**(1/3) - x + 3*log(x**(1/3) + 1)

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GIAC/XCAS [A]  time = 0.223957, size = 36, normalized size = 0.92 \[ \frac{3}{4} \, x^{\frac{4}{3}} - x + \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 3 \,{\rm ln}\left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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