Optimal. Leaf size=242 \[ -\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac{1}{2}\right )}{8\ 2^{3/4} x}-\frac{9 \sqrt [4]{3 x^2-2} x}{8 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{3 \left (3 x^2-2\right )^{3/4}}{8 x}+\frac{\left (3 x^2-2\right )^{3/4}}{6 x^3}+\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{4\ 2^{3/4} x} \]
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Rubi [A] time = 0.109254, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {325, 230, 305, 220, 1196} \[ -\frac{9 \sqrt [4]{3 x^2-2} x}{8 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{3 \left (3 x^2-2\right )^{3/4}}{8 x}+\frac{\left (3 x^2-2\right )^{3/4}}{6 x^3}-\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{8\ 2^{3/4} x}+\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{4\ 2^{3/4} x} \]
Antiderivative was successfully verified.
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Rule 325
Rule 230
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt [4]{-2+3 x^2}} \, dx &=\frac{\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac{3}{4} \int \frac{1}{x^2 \sqrt [4]{-2+3 x^2}} \, dx\\ &=\frac{\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac{3 \left (-2+3 x^2\right )^{3/4}}{8 x}-\frac{9}{16} \int \frac{1}{\sqrt [4]{-2+3 x^2}} \, dx\\ &=\frac{\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac{3 \left (-2+3 x^2\right )^{3/4}}{8 x}-\frac{\left (3 \sqrt{\frac{3}{2}} \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{8 x}\\ &=\frac{\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac{3 \left (-2+3 x^2\right )^{3/4}}{8 x}-\frac{\left (3 \sqrt{3} \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{8 x}+\frac{\left (3 \sqrt{3} \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{x^2}{\sqrt{2}}}{\sqrt{1+\frac{x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{8 x}\\ &=\frac{\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac{3 \left (-2+3 x^2\right )^{3/4}}{8 x}-\frac{9 x \sqrt [4]{-2+3 x^2}}{8 \left (\sqrt{2}+\sqrt{-2+3 x^2}\right )}+\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{2}+\sqrt{-2+3 x^2}\right )^2}} \left (\sqrt{2}+\sqrt{-2+3 x^2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{4\ 2^{3/4} x}-\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{2}+\sqrt{-2+3 x^2}\right )^2}} \left (\sqrt{2}+\sqrt{-2+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{8\ 2^{3/4} x}\\ \end{align*}
Mathematica [C] time = 0.0072733, size = 48, normalized size = 0.2 \[ -\frac{\sqrt [4]{1-\frac{3 x^2}{2}} \, _2F_1\left (-\frac{3}{2},\frac{1}{4};-\frac{1}{2};\frac{3 x^2}{2}\right )}{3 x^3 \sqrt [4]{3 x^2-2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.037, size = 67, normalized size = 0.3 \begin{align*}{\frac{27\,{x}^{4}-6\,{x}^{2}-8}{24\,{x}^{3}}{\frac{1}{\sqrt [4]{3\,{x}^{2}-2}}}}-{\frac{9\,{2}^{3/4}x}{32}\sqrt [4]{-{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}{\frac{1}{\sqrt [4]{{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{1}{4}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (3 \, x^{2} - 2\right )}^{\frac{3}{4}}}{3 \, x^{6} - 2 \, x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.82427, size = 34, normalized size = 0.14 \begin{align*} \frac{2^{\frac{3}{4}} e^{\frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{1}{4}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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