Optimal. Leaf size=63 \[ \frac{3 x}{2 \sqrt [4]{3 x^2+2}}-\frac{\left (3 x^2+2\right )^{3/4}}{2 x}-\frac{\sqrt{3} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2^{3/4}} \]
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Rubi [A] time = 0.0127797, antiderivative size = 63, 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, 227, 196} \[ \frac{3 x}{2 \sqrt [4]{3 x^2+2}}-\frac{\left (3 x^2+2\right )^{3/4}}{2 x}-\frac{\sqrt{3} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2^{3/4}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt [4]{2+3 x^2}} \, dx &=-\frac{\left (2+3 x^2\right )^{3/4}}{2 x}+\frac{3}{4} \int \frac{1}{\sqrt [4]{2+3 x^2}} \, dx\\ &=\frac{3 x}{2 \sqrt [4]{2+3 x^2}}-\frac{\left (2+3 x^2\right )^{3/4}}{2 x}-\frac{3}{2} \int \frac{1}{\left (2+3 x^2\right )^{5/4}} \, dx\\ &=\frac{3 x}{2 \sqrt [4]{2+3 x^2}}-\frac{\left (2+3 x^2\right )^{3/4}}{2 x}-\frac{\sqrt{3} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0054375, size = 27, normalized size = 0.43 \[ -\frac{\, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{3 x^2}{2}\right )}{\sqrt [4]{2} x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 33, normalized size = 0.5 \begin{align*} -{\frac{1}{2\,x} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{4}}}}+{\frac{3\,{2}^{3/4}x}{8}{\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}} 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{3}{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 = 0.664222, size = 29, normalized size = 0.46 \begin{align*} - \frac{2^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{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{1}{4}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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