Optimal. Leaf size=48 \[ \frac{2}{27} \left (3 x^2-1\right )^{3/4}+\frac{2}{9} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{2}{9} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
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Rubi [A] time = 0.032934, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 80, 63, 298, 203, 206} \[ \frac{2}{27} \left (3 x^2-1\right )^{3/4}+\frac{2}{9} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{2}{9} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 63
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(-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{1}{3} \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{4}{9} \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}{9} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac{2}{9} \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}{9} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac{2}{9} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0101469, size = 44, normalized size = 0.92 \[ \frac{2}{27} \left (\left (3 x^2-1\right )^{3/4}+3 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-3 \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{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 = 1.51309, size = 70, normalized size = 1.46 \begin{align*} \frac{2}{27} \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} + \frac{2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{9} \, \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 = 1.32912, size = 165, normalized size = 3.44 \begin{align*} \frac{2}{27} \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} + \frac{2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{9} \, \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 = 10.2749, size = 58, normalized size = 1.21 \begin{align*} \frac{2 \left (3 x^{2} - 1\right )^{\frac{3}{4}}}{27} + \frac{\log{\left (\sqrt [4]{3 x^{2} - 1} - 1 \right )}}{9} - \frac{\log{\left (\sqrt [4]{3 x^{2} - 1} + 1 \right )}}{9} + \frac{2 \operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23998, size = 72, normalized size = 1.5 \begin{align*} \frac{2}{27} \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} + \frac{2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{9} \, \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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