Optimal. Leaf size=43 \[ \frac{2 x}{\sqrt [4]{3 x^2+2}}-\frac{2 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0069099, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {227, 196} \[ \frac{2 x}{\sqrt [4]{3 x^2+2}}-\frac{2 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{2+3 x^2}} \, dx &=\frac{2 x}{\sqrt [4]{2+3 x^2}}-2 \int \frac{1}{\left (2+3 x^2\right )^{5/4}} \, dx\\ &=\frac{2 x}{\sqrt [4]{2+3 x^2}}-\frac{2 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0033747, size = 24, normalized size = 0.56 \[ \frac{x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right )}{\sqrt [4]{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 18, normalized size = 0.4 \begin{align*}{\frac{{2}^{{\frac{3}{4}}}x}{2}{\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}}}\,{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{1}{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.58462, size = 26, normalized size = 0.6 \begin{align*} \frac{2^{\frac{3}{4}} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \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{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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