Optimal. Leaf size=33 \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]
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Rubi [A] time = 0.0190299, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2012, 1593, 266, 43} \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 2012
Rule 1593
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{-\sqrt{1+x}+(1+x)^{2/3}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{-\sqrt{x}+x^{2/3}} \, dx,x,1+x\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{\left (-1+\sqrt [6]{x}\right ) \sqrt{x}} \, dx,x,1+x\right )\\ &=6 \operatorname{Subst}\left (\int \frac{x^2}{-1+x} \, dx,x,\sqrt [6]{1+x}\right )\\ &=6 \operatorname{Subst}\left (\int \left (1+\frac{1}{-1+x}+x\right ) \, dx,x,\sqrt [6]{1+x}\right )\\ &=6 \sqrt [6]{1+x}+3 \sqrt [3]{1+x}+6 \log \left (1-\sqrt [6]{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0214325, size = 33, normalized size = 1. \[ 3 \left (\sqrt [3]{x+1}+2 \sqrt [6]{x+1}+2 \log \left (1-\sqrt [6]{x+1}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.031, size = 111, normalized size = 3.4 \begin{align*} 6\,\sqrt [6]{1+x}+3\,\sqrt [3]{1+x}+\ln \left ( x \right ) +2\,\ln \left ( -1+\sqrt [6]{1+x} \right ) -\ln \left ( \sqrt [3]{1+x}+\sqrt [6]{1+x}+1 \right ) -2\,\ln \left ( 1+\sqrt [6]{1+x} \right ) +\ln \left ( \sqrt [3]{1+x}-\sqrt [6]{1+x}+1 \right ) -\ln \left ( 1+\sqrt{1+x} \right ) +\ln \left ( -1+\sqrt{1+x} \right ) +2\,\ln \left ( -1+\sqrt [3]{1+x} \right ) -\ln \left ( \left ( 1+x \right ) ^{{\frac{2}{3}}}+\sqrt [3]{1+x}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.927302, size = 34, normalized size = 1.03 \begin{align*} 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 6 \,{\left (x + 1\right )}^{\frac{1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac{1}{6}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72372, size = 84, normalized size = 2.55 \begin{align*} 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 6 \,{\left (x + 1\right )}^{\frac{1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\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}{\left (x + 1\right )^{\frac{2}{3}} - \sqrt{x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0838, size = 35, normalized size = 1.06 \begin{align*} 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 6 \,{\left (x + 1\right )}^{\frac{1}{6}} + 6 \, \log \left ({\left |{\left (x + 1\right )}^{\frac{1}{6}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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