Optimal. Leaf size=49 \[ -\frac{\sqrt{3} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{2 \sqrt [4]{2}}-\frac{\sqrt [4]{3 x^2+2}}{2 x} \]
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Rubi [A] time = 0.0083085, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {325, 231} \[ -\frac{\sqrt [4]{3 x^2+2}}{2 x}-\frac{\sqrt{3} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 231
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{\sqrt{3} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}}\\ \end{align*}
Mathematica [C] time = 0.0047161, size = 27, normalized size = 0.55 \[ -\frac{\, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{1}{2};-\frac{3 x^2}{2}\right )}{2^{3/4} x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 33, normalized size = 0.7 \begin{align*} -{\frac{1}{2\,x}\sqrt [4]{3\,{x}^{2}+2}}-{\frac{3\,\sqrt [4]{2}x}{8}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\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{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*}{\rm integral}\left (\frac{{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}}}{3 \, x^{4} + 2 \, x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.768287, size = 29, normalized size = 0.59 \begin{align*} - \frac{\sqrt [4]{2}{{}_{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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