Optimal. Leaf size=141 \[ \frac{27 \sqrt{3} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac{1}{2}\right )}{64 \sqrt [4]{2} x}+\frac{27 \sqrt [4]{-3 x^2-2}}{32 x}-\frac{9 \sqrt [4]{-3 x^2-2}}{40 x^3}+\frac{\sqrt [4]{-3 x^2-2}}{10 x^5} \]
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Rubi [A] time = 0.057696, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {325, 234, 220} \[ \frac{27 \sqrt [4]{-3 x^2-2}}{32 x}-\frac{9 \sqrt [4]{-3 x^2-2}}{40 x^3}+\frac{\sqrt [4]{-3 x^2-2}}{10 x^5}+\frac{27 \sqrt{3} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{64 \sqrt [4]{2} x} \]
Antiderivative was successfully verified.
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Rule 325
Rule 234
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (-2-3 x^2\right )^{3/4}} \, dx &=\frac{\sqrt [4]{-2-3 x^2}}{10 x^5}-\frac{27}{20} \int \frac{1}{x^4 \left (-2-3 x^2\right )^{3/4}} \, dx\\ &=\frac{\sqrt [4]{-2-3 x^2}}{10 x^5}-\frac{9 \sqrt [4]{-2-3 x^2}}{40 x^3}+\frac{27}{16} \int \frac{1}{x^2 \left (-2-3 x^2\right )^{3/4}} \, dx\\ &=\frac{\sqrt [4]{-2-3 x^2}}{10 x^5}-\frac{9 \sqrt [4]{-2-3 x^2}}{40 x^3}+\frac{27 \sqrt [4]{-2-3 x^2}}{32 x}-\frac{81}{64} \int \frac{1}{\left (-2-3 x^2\right )^{3/4}} \, dx\\ &=\frac{\sqrt [4]{-2-3 x^2}}{10 x^5}-\frac{9 \sqrt [4]{-2-3 x^2}}{40 x^3}+\frac{27 \sqrt [4]{-2-3 x^2}}{32 x}+\frac{\left (27 \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 )}{32 x}\\ &=\frac{\sqrt [4]{-2-3 x^2}}{10 x^5}-\frac{9 \sqrt [4]{-2-3 x^2}}{40 x^3}+\frac{27 \sqrt [4]{-2-3 x^2}}{32 x}+\frac{27 \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 )}{64 \sqrt [4]{2} x}\\ \end{align*}
Mathematica [C] time = 0.0079166, size = 48, normalized size = 0.34 \[ -\frac{\left (\frac{3 x^2}{2}+1\right )^{3/4} \, _2F_1\left (-\frac{5}{2},\frac{3}{4};-\frac{3}{2};-\frac{3 x^2}{2}\right )}{5 x^5 \left (-3 x^2-2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}} \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^{6}}\,{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{160 \, x^{5}{\rm integral}\left (\frac{81 \,{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}{64 \,{\left (3 \, x^{2} + 2\right )}}, x\right ) +{\left (135 \, x^{4} - 36 \, x^{2} + 16\right )}{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}{160 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.14833, size = 37, normalized size = 0.26 \begin{align*} \frac{\sqrt [4]{2} e^{\frac{i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{3}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{10 x^{5}} \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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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