3.407 \(\int \frac{1}{\sqrt [3]{x}+\sqrt{x}} \, dx\)

Optimal. Leaf size=32 \[ 2 \sqrt{x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \]

[Out]

6*x^(1/6) - 3*x^(1/3) + 2*Sqrt[x] - 6*Log[1 + x^(1/6)]

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Rubi [A]  time = 0.0298855, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ 2 \sqrt{x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x^(1/3) + Sqrt[x])^(-1),x]

[Out]

6*x^(1/6) - 3*x^(1/3) + 2*Sqrt[x] - 6*Log[1 + x^(1/6)]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 6 \sqrt [6]{x} + 2 \sqrt{x} - 6 \log{\left (\sqrt [6]{x} + 1 \right )} - 6 \int ^{\sqrt [6]{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*x**(1/6) + 2*sqrt(x) - 6*log(x**(1/6) + 1) - 6*Integral(x, (x, x**(1/6)))

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Mathematica [A]  time = 0.0134489, size = 32, normalized size = 1. \[ 2 \sqrt{x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

6*x^(1/6) - 3*x^(1/3) + 2*Sqrt[x] - 6*Log[1 + x^(1/6)]

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Maple [B]  time = 0.037, size = 92, normalized size = 2.9 \[ -\ln \left ( \sqrt [3]{x}+\sqrt [6]{x}+1 \right ) +2\,\ln \left ( \sqrt [6]{x}-1 \right ) -2\,\ln \left ( 1+\sqrt [6]{x} \right ) +\ln \left ( 1-\sqrt [6]{x}+\sqrt [3]{x} \right ) +2\,\sqrt{x}+\ln \left ( -1+\sqrt{x} \right ) -\ln \left ( 1+\sqrt{x} \right ) +6\,\sqrt [6]{x}-\ln \left ( -1+x \right ) +\ln \left ({x}^{{\frac{2}{3}}}+\sqrt [3]{x}+1 \right ) -2\,\ln \left ( \sqrt [3]{x}-1 \right ) -3\,\sqrt [3]{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^(1/3)+x^(1/2)),x)

[Out]

-ln(x^(1/3)+x^(1/6)+1)+2*ln(x^(1/6)-1)-2*ln(1+x^(1/6))+ln(1-x^(1/6)+x^(1/3))+2*x
^(1/2)+ln(-1+x^(1/2))-ln(1+x^(1/2))+6*x^(1/6)-ln(-1+x)+ln(x^(2/3)+x^(1/3)+1)-2*l
n(x^(1/3)-1)-3*x^(1/3)

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Maxima [A]  time = 0.694591, size = 32, normalized size = 1. \[ 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \log \left (x^{\frac{1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) - 3*x^(1/3) + 6*x^(1/6) - 6*log(x^(1/6) + 1)

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Fricas [A]  time = 0.268479, size = 32, normalized size = 1. \[ 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \log \left (x^{\frac{1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) - 3*x^(1/3) + 6*x^(1/6) - 6*log(x^(1/6) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{x} + \sqrt{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**(1/3) + sqrt(x)), x)

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GIAC/XCAS [A]  time = 0.277637, size = 32, normalized size = 1. \[ 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \,{\rm ln}\left (x^{\frac{1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) - 3*x^(1/3) + 6*x^(1/6) - 6*ln(x^(1/6) + 1)