Optimal. Leaf size=32 \[ 6 \sqrt [6]{x}-3 \sqrt [3]{x}+2 \sqrt {x}-6 \log \left (1+\sqrt [6]{x}\right ) \]
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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 = {1607, 272, 45}
\begin {gather*} 2 \sqrt {x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1607
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 \text {Subst}\left (\int \frac {x^3}{1+x} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \text {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 = 0.00, size = 33, normalized size = 1.03 \begin {gather*} \left (6-3 \sqrt [6]{x}+2 \sqrt [3]{x}\right ) \sqrt [6]{x}-6 \log \left (1+\sqrt [6]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs.
\(2(24)=48\).
time = 0.03, size = 92, normalized size = 2.88
method | result | size |
derivativedivides | \(6 x^{\frac {1}{6}}-3 x^{\frac {1}{3}}-6 \ln \left (1+x^{\frac {1}{6}}\right )+2 \sqrt {x}\) | \(25\) |
meijerg | \(\frac {x^{\frac {1}{6}} \left (4 x^{\frac {1}{3}}-6 x^{\frac {1}{6}}+12\right )}{2}-6 \ln \left (1+x^{\frac {1}{6}}\right )\) | \(27\) |
default | \(2 \ln \left (-1+x^{\frac {1}{6}}\right )-\ln \left (x^{\frac {1}{3}}+x^{\frac {1}{6}}+1\right )-2 \ln \left (1+x^{\frac {1}{6}}\right )+\ln \left (x^{\frac {1}{3}}-x^{\frac {1}{6}}+1\right )+2 \sqrt {x}+\ln \left (\sqrt {x}-1\right )-\ln \left (1+\sqrt {x}\right )+6 x^{\frac {1}{6}}-\ln \left (-1+x \right )-2 \ln \left (-1+x^{\frac {1}{3}}\right )+\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )-3 x^{\frac {1}{3}}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.27, size = 24, normalized size = 0.75 \begin {gather*} 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \log \left (x^{\frac {1}{6}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 24, normalized size = 0.75 \begin {gather*} 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \log \left (x^{\frac {1}{6}} + 1\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt [3]{x} + \sqrt {x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 24, normalized size = 0.75 \begin {gather*} 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \log \left (x^{\frac {1}{6}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 24, normalized size = 0.75 \begin {gather*} 2\,\sqrt {x}-6\,\ln \left (x^{1/6}+1\right )-3\,x^{1/3}+6\,x^{1/6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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