Optimal. Leaf size=105 \[ \frac{\sqrt{3} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac{1}{2}\right )}{4 \sqrt [4]{2} x}+\frac{\sqrt [4]{-3 x^2-2}}{2 x} \]
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Rubi [A] time = 0.0353193, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {325, 234, 220} \[ \frac{\sqrt [4]{-3 x^2-2}}{2 x}+\frac{\sqrt{3} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{2} x} \]
Antiderivative was successfully verified.
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Rule 325
Rule 234
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (-2-3 x^2\right )^{3/4}} \, dx &=\frac{\sqrt [4]{-2-3 x^2}}{2 x}-\frac{3}{4} \int \frac{1}{\left (-2-3 x^2\right )^{3/4}} \, dx\\ &=\frac{\sqrt [4]{-2-3 x^2}}{2 x}+\frac{\left (\sqrt{\frac{3}{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 )}{2 x}\\ &=\frac{\sqrt [4]{-2-3 x^2}}{2 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 )}{4 \sqrt [4]{2} x}\\ \end{align*}
Mathematica [C] time = 0.0067861, size = 46, normalized size = 0.44 \[ -\frac{\left (\frac{3 x^2}{2}+1\right )^{3/4} \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{1}{2};-\frac{3 x^2}{2}\right )}{x \left (-3 x^2-2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( -3\,{x}^{2}-2 \right ) ^{-{\frac{3}{4}}}}\, dx \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{3}{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{1}{4}}}{4 \,{\left (3 \, x^{2} + 2\right )}}, x\right ) +{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.754902, size = 34, normalized size = 0.32 \begin{align*} \frac{\sqrt [4]{2} e^{\frac{i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{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{3}{4}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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