Optimal. Leaf size=202 \[ -\frac{\sqrt [4]{2} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac{1}{2}\right )}{\sqrt{3} x}+\frac{2 \sqrt [4]{-3 x^2-2} x}{\sqrt{-3 x^2-2}+\sqrt{2}}+\frac{2 \sqrt [4]{2} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{\sqrt{3} x} \]
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Rubi [A] time = 0.0784781, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {230, 305, 220, 1196} \[ \frac{2 \sqrt [4]{-3 x^2-2} x}{\sqrt{-3 x^2-2}+\sqrt{2}}-\frac{\sqrt [4]{2} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{\sqrt{3} x}+\frac{2 \sqrt [4]{2} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{\sqrt{3} x} \]
Antiderivative was successfully verified.
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Rule 230
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{-2-3 x^2}} \, dx &=-\frac{\left (\sqrt{\frac{2}{3}} \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 )}{x}\\ &=-\frac{\left (2 \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 )}{\sqrt{3} x}+\frac{\left (2 \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 )}{\sqrt{3} x}\\ &=\frac{2 x \sqrt [4]{-2-3 x^2}}{\sqrt{2}+\sqrt{-2-3 x^2}}+\frac{2 \sqrt [4]{2} \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 )}{\sqrt{3} x}-\frac{\sqrt [4]{2} \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 )}{\sqrt{3} x}\\ \end{align*}
Mathematica [C] time = 0.0060325, size = 43, normalized size = 0.21 \[ \frac{x \sqrt [4]{\frac{3 x^2}{2}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right )}{\sqrt [4]{-3 x^2-2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 21, normalized size = 0.1 \begin{align*} -{\frac{ \left ( -1 \right ) ^{{\frac{3}{4}}}x{2}^{{\frac{3}{4}}}}{2}{\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}}}\,{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{3 \, x{\rm integral}\left (-\frac{4 \,{\left (-3 \, x^{2} - 2\right )}^{\frac{3}{4}}}{3 \,{\left (3 \, x^{4} + 2 \, x^{2}\right )}}, x\right ) - 2 \,{\left (-3 \, x^{2} - 2\right )}^{\frac{3}{4}}}{3 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.598373, size = 32, normalized size = 0.16 \begin{align*} \frac{2^{\frac{3}{4}} x e^{- \frac{i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{2} \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}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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