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 = 0.0142512, 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, 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 1593
Rule 266
Rule 43
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*}
Mathematica [A] time = 0.0119524, 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.
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Maple [B] time = 0.023, size = 92, normalized size = 2.9 \begin{align*} 2\,\ln \left ( \sqrt [6]{x}-1 \right ) -\ln \left ( \sqrt [3]{x}+\sqrt [6]{x}+1 \right ) +\ln \left ( 1-\sqrt [6]{x}+\sqrt [3]{x} \right ) -2\,\ln \left ( 1+\sqrt [6]{x} \right ) +2\,\sqrt{x}+\ln \left ( -1+\sqrt{x} \right ) -\ln \left ( 1+\sqrt{x} \right ) +6\,\sqrt [6]{x}-\ln \left ( x-1 \right ) -2\,\ln \left ( \sqrt [3]{x}-1 \right ) +\ln \left ({x}^{{\frac{2}{3}}}+\sqrt [3]{x}+1 \right ) -3\,\sqrt [3]{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26138, size = 32, normalized size = 1. \begin{align*} 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24838, size = 76, normalized size = 2.38 \begin{align*} 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{x} + \sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12005, size = 32, normalized size = 1. \begin{align*} 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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