Optimal. Leaf size=14 \[ \frac{5}{2} \log \left (1-x^{2/5}\right ) \]
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Rubi [A] time = 0.0046696, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 260} \[ \frac{5}{2} \log \left (1-x^{2/5}\right ) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{-x^{3/5}+x} \, dx &=\int \frac{1}{\left (-1+x^{2/5}\right ) x^{3/5}} \, dx\\ &=\frac{5}{2} \log \left (1-x^{2/5}\right )\\ \end{align*}
Mathematica [A] time = 0.0024372, size = 14, normalized size = 1. \[ \frac{5}{2} \log \left (1-x^{2/5}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.096, size = 116, normalized size = 8.3 \begin{align*}{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{1}{2}\ln \left ( -\sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}+\sqrt [5]{x}+2 \right ) }-{\frac{1}{2}\ln \left ( \sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}+\sqrt [5]{x}+2 \right ) }+2\,\ln \left ( -1+\sqrt [5]{x} \right ) +2\,\ln \left ( 1+\sqrt [5]{x} \right ) -{\frac{1}{2}\ln \left ( -\sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}-\sqrt [5]{x}+2 \right ) }-{\frac{1}{2}\ln \left ( \sqrt{5}\sqrt [5]{x}+2\,{x}^{2/5}-\sqrt [5]{x}+2 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0348, size = 23, normalized size = 1.64 \begin{align*} \frac{5}{2} \, \log \left (x^{\frac{1}{5}} + 1\right ) + \frac{5}{2} \, \log \left (x^{\frac{1}{5}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.787572, size = 30, normalized size = 2.14 \begin{align*} \frac{5}{2} \, \log \left (x^{\frac{2}{5}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.385201, size = 22, normalized size = 1.57 \begin{align*} \frac{5 \log{\left (\sqrt [5]{x} - 1 \right )}}{2} + \frac{5 \log{\left (\sqrt [5]{x} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17703, size = 24, normalized size = 1.71 \begin{align*} \frac{5}{2} \, \log \left (x^{\frac{1}{5}} + 1\right ) + \frac{5}{2} \, \log \left ({\left | x^{\frac{1}{5}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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