Optimal. Leaf size=32 \[ 2 \sqrt {x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1593, 266, 43} \[ 2 \sqrt {x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{x}+\sqrt {x}} \, dx &=\int \frac {1}{\left (1+\sqrt [6]{x}\right ) \sqrt [3]{x}} \, dx\\ &=6 \operatorname {Subst}\left (\int \frac {x^3}{1+x} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname {Subst}\left (\int \left (1+\frac {1}{-1-x}-x+x^2\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}-3 \sqrt [3]{x}+2 \sqrt {x}-6 \log \left (1+\sqrt [6]{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ 2 \sqrt {x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 24, normalized size = 0.75 \[ 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \log \left (x^{\frac {1}{6}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 24, normalized size = 0.75 \[ 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \log \left (x^{\frac {1}{6}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 92, normalized size = 2.88 \[ -\ln \left (x -1\right )+\ln \left (\sqrt {x}-1\right )-\ln \left (\sqrt {x}+1\right )-2 \ln \left (x^{\frac {1}{6}}+1\right )-2 \ln \left (x^{\frac {1}{3}}-1\right )+2 \ln \left (x^{\frac {1}{6}}-1\right )+\ln \left (x^{\frac {1}{3}}-x^{\frac {1}{6}}+1\right )-\ln \left (x^{\frac {1}{3}}+x^{\frac {1}{6}}+1\right )+\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )+2 \sqrt {x}-3 x^{\frac {1}{3}}+6 x^{\frac {1}{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 24, normalized size = 0.75 \[ 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \log \left (x^{\frac {1}{6}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 0.75 \[ 2\,\sqrt {x}-6\,\ln \left (x^{1/6}+1\right )-3\,x^{1/3}+6\,x^{1/6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{x} + \sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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