Optimal. Leaf size=27 \[ \frac{2^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0029293, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {231} \[ \frac{2^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 231
Rubi steps
\begin{align*} \int \frac{1}{\left (2+3 x^2\right )^{3/4}} \, dx &=\frac{2^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.003973, size = 24, normalized size = 0.89 \[ \frac{x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right )}{2^{3/4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 18, normalized size = 0.7 \begin{align*}{\frac{\sqrt [4]{2}x}{2}{\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}}}\,{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{1}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.588789, size = 26, normalized size = 0.96 \begin{align*} \frac{\sqrt [4]{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \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{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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