Optimal. Leaf size=59 \[ \frac{1}{42} \log \left (\sqrt [3]{2 x^7-27}+3\right )-\frac{\tan ^{-1}\left (\frac{3-2 \sqrt [3]{2 x^7-27}}{3 \sqrt{3}}\right )}{21 \sqrt{3}}-\frac{\log (x)}{18} \]
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Rubi [A] time = 0.0371693, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 58, 618, 204, 31} \[ \frac{1}{42} \log \left (\sqrt [3]{2 x^7-27}+3\right )-\frac{\tan ^{-1}\left (\frac{3-2 \sqrt [3]{2 x^7-27}}{3 \sqrt{3}}\right )}{21 \sqrt{3}}-\frac{\log (x)}{18} \]
Antiderivative was successfully verified.
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Rule 266
Rule 58
Rule 618
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \left (-27+2 x^7\right )^{2/3}} \, dx &=\frac{1}{7} \operatorname{Subst}\left (\int \frac{1}{x (-27+2 x)^{2/3}} \, dx,x,x^7\right )\\ &=-\frac{\log (x)}{18}+\frac{1}{42} \operatorname{Subst}\left (\int \frac{1}{3+x} \, dx,x,\sqrt [3]{-27+2 x^7}\right )+\frac{1}{14} \operatorname{Subst}\left (\int \frac{1}{9-3 x+x^2} \, dx,x,\sqrt [3]{-27+2 x^7}\right )\\ &=-\frac{\log (x)}{18}+\frac{1}{42} \log \left (3+\sqrt [3]{-27+2 x^7}\right )-\frac{1}{7} \operatorname{Subst}\left (\int \frac{1}{-27-x^2} \, dx,x,-3+2 \sqrt [3]{-27+2 x^7}\right )\\ &=-\frac{\tan ^{-1}\left (\frac{3-2 \sqrt [3]{-27+2 x^7}}{3 \sqrt{3}}\right )}{21 \sqrt{3}}-\frac{\log (x)}{18}+\frac{1}{42} \log \left (3+\sqrt [3]{-27+2 x^7}\right )\\ \end{align*}
Mathematica [A] time = 0.0221199, size = 84, normalized size = 1.42 \[ \frac{1}{63} \log \left (\sqrt [3]{2 x^7-27}+3\right )-\frac{1}{126} \log \left (\left (2 x^7-27\right )^{2/3}-3 \sqrt [3]{2 x^7-27}+9\right )+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{2 x^7-27}-3}{3 \sqrt{3}}\right )}{21 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.058, size = 74, normalized size = 1.3 \begin{align*}{\frac{1}{63\,\Gamma \left ( 2/3 \right ) } \left ( -{\it signum} \left ( -1+{\frac{2\,{x}^{7}}{27}} \right ) \right ) ^{{\frac{2}{3}}} \left ( \left ({\frac{\pi \,\sqrt{3}}{6}}-{\frac{9\,\ln \left ( 3 \right ) }{2}}+7\,\ln \left ( x \right ) +\ln \left ( 2 \right ) +i\pi \right ) \Gamma \left ({\frac{2}{3}} \right ) +{\frac{4\,\Gamma \left ( 2/3 \right ){x}^{7}}{81}{\mbox{$_3$F$_2$}(1,1,{\frac{5}{3}};\,2,2;\,{\frac{2\,{x}^{7}}{27}})}} \right ) \left ({\it signum} \left ( -1+{\frac{2\,{x}^{7}}{27}} \right ) \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4436, size = 86, normalized size = 1.46 \begin{align*} \frac{1}{63} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (2 \,{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} - 3\right )}\right ) - \frac{1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac{2}{3}} - 3 \,{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 9\right ) + \frac{1}{63} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94792, size = 217, normalized size = 3.68 \begin{align*} \frac{1}{63} \, \sqrt{3} \arctan \left (\frac{2}{9} \, \sqrt{3}{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) - \frac{1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac{2}{3}} - 3 \,{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 9\right ) + \frac{1}{63} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.03546, size = 42, normalized size = 0.71 \begin{align*} - \frac{\sqrt [3]{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{27 e^{2 i \pi }}{2 x^{7}}} \right )}}{14 x^{\frac{14}{3}} \Gamma \left (\frac{5}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09344, size = 86, normalized size = 1.46 \begin{align*} \frac{1}{63} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (2 \,{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} - 3\right )}\right ) - \frac{1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac{2}{3}} - 3 \,{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 9\right ) + \frac{1}{63} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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