Optimal. Leaf size=63 \[ \frac{2}{189} \left (3 x^2-1\right )^{7/4}+\frac{2}{27} \left (3 x^2-1\right )^{3/4}+\frac{4}{27} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{4}{27} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
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Rubi [A] time = 0.0480356, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 88, 63, 298, 203, 206} \[ \frac{2}{189} \left (3 x^2-1\right )^{7/4}+\frac{2}{27} \left (3 x^2-1\right )^{3/4}+\frac{4}{27} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{4}{27} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rule 63
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{3 \sqrt [4]{-1+3 x}}+\frac{4}{9 (-2+3 x) \sqrt [4]{-1+3 x}}+\frac{1}{9} (-1+3 x)^{3/4}\right ) \, dx,x,x^2\right )\\ &=\frac{2}{27} \left (-1+3 x^2\right )^{3/4}+\frac{2}{189} \left (-1+3 x^2\right )^{7/4}+\frac{2}{9} \operatorname{Subst}\left (\int \frac{1}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )\\ &=\frac{2}{27} \left (-1+3 x^2\right )^{3/4}+\frac{2}{189} \left (-1+3 x^2\right )^{7/4}+\frac{8}{27} \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=\frac{2}{27} \left (-1+3 x^2\right )^{3/4}+\frac{2}{189} \left (-1+3 x^2\right )^{7/4}-\frac{4}{27} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac{4}{27} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=\frac{2}{27} \left (-1+3 x^2\right )^{3/4}+\frac{2}{189} \left (-1+3 x^2\right )^{7/4}+\frac{4}{27} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac{4}{27} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0382267, size = 51, normalized size = 0.81 \[ \frac{2}{189} \left (3 \left (3 x^2-1\right )^{3/4} \left (x^2+2\right )+14 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-14 \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{3\,{x}^{2}-2}{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44819, size = 85, normalized size = 1.35 \begin{align*} \frac{2}{189} \,{\left (3 \, x^{2} - 1\right )}^{\frac{7}{4}} + \frac{2}{27} \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} + \frac{4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33612, size = 182, normalized size = 2.89 \begin{align*} \frac{2}{63} \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}{\left (x^{2} + 2\right )} + \frac{4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.9766, size = 75, normalized size = 1.19 \begin{align*} \frac{2 \left (3 x^{2} - 1\right )^{\frac{7}{4}}}{189} + \frac{2 \left (3 x^{2} - 1\right )^{\frac{3}{4}}}{27} + \frac{2 \log{\left (\sqrt [4]{3 x^{2} - 1} - 1 \right )}}{27} - \frac{2 \log{\left (\sqrt [4]{3 x^{2} - 1} + 1 \right )}}{27} + \frac{4 \operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22778, size = 86, normalized size = 1.37 \begin{align*} \frac{2}{189} \,{\left (3 \, x^{2} - 1\right )}^{\frac{7}{4}} + \frac{2}{27} \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} + \frac{4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{2}{27} \, \log \left ({\left |{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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