Optimal. Leaf size=223 \[ -\frac{\sqrt{3} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac{1}{2}\right )}{2\ 2^{3/4} x}+\frac{3 \sqrt [4]{-3 x^2-2} x}{2 \left (\sqrt{-3 x^2-2}+\sqrt{2}\right )}+\frac{\left (-3 x^2-2\right )^{3/4}}{2 x}+\frac{\sqrt{3} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{2^{3/4} x} \]
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Rubi [A] time = 0.092718, antiderivative size = 223, 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 = {325, 230, 305, 220, 1196} \[ \frac{3 \sqrt [4]{-3 x^2-2} x}{2 \left (\sqrt{-3 x^2-2}+\sqrt{2}\right )}+\frac{\left (-3 x^2-2\right )^{3/4}}{2 x}-\frac{\sqrt{3} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{2\ 2^{3/4} x}+\frac{\sqrt{3} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{2^{3/4} x} \]
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^2 \sqrt [4]{-2-3 x^2}} \, dx &=\frac{\left (-2-3 x^2\right )^{3/4}}{2 x}+\frac{3}{4} \int \frac{1}{\sqrt [4]{-2-3 x^2}} \, dx\\ &=\frac{\left (-2-3 x^2\right )^{3/4}}{2 x}-\frac{\left (\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 )}{2 x}\\ &=\frac{\left (-2-3 x^2\right )^{3/4}}{2 x}-\frac{\left (\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 )}{2 x}+\frac{\left (\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 )}{2 x}\\ &=\frac{\left (-2-3 x^2\right )^{3/4}}{2 x}+\frac{3 x \sqrt [4]{-2-3 x^2}}{2 \left (\sqrt{2}+\sqrt{-2-3 x^2}\right )}+\frac{\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 )}{2^{3/4} x}-\frac{\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 )}{2\ 2^{3/4} x}\\ \end{align*}
Mathematica [C] time = 0.0074077, size = 46, normalized size = 0.21 \[ -\frac{\sqrt [4]{\frac{3 x^2}{2}+1} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{3 x^2}{2}\right )}{x \sqrt [4]{-3 x^2-2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 43, normalized size = 0.2 \begin{align*} -{\frac{3\,{x}^{2}+2}{2\,x}{\frac{1}{\sqrt [4]{-3\,{x}^{2}-2}}}}-{\frac{3\, \left ( -1 \right ) ^{3/4}x{2}^{3/4}}{8}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{\frac{3\,{x}^{2}}{2}})}} \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^{2}}\,{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*} \frac{2 \, x{\rm integral}\left (-\frac{3 \,{\left (-3 \, x^{2} - 2\right )}^{\frac{3}{4}}}{4 \,{\left (3 \, x^{2} + 2\right )}}, x\right ) +{\left (-3 \, x^{2} - 2\right )}^{\frac{3}{4}}}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.669019, size = 36, normalized size = 0.16 \begin{align*} \frac{2^{\frac{3}{4}} e^{\frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{2 x} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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