Optimal. Leaf size=63 \[ \frac{2}{15} \left (3 x^2+2\right )^{3/4} x-\frac{8 x}{15 \sqrt [4]{3 x^2+2}}+\frac{8 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{15 \sqrt{3}} \]
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Rubi [A] time = 0.012527, 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 = {321, 227, 196} \[ \frac{2}{15} \left (3 x^2+2\right )^{3/4} x-\frac{8 x}{15 \sqrt [4]{3 x^2+2}}+\frac{8 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{15 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt [4]{2+3 x^2}} \, dx &=\frac{2}{15} x \left (2+3 x^2\right )^{3/4}-\frac{4}{15} \int \frac{1}{\sqrt [4]{2+3 x^2}} \, dx\\ &=-\frac{8 x}{15 \sqrt [4]{2+3 x^2}}+\frac{2}{15} x \left (2+3 x^2\right )^{3/4}+\frac{8}{15} \int \frac{1}{\left (2+3 x^2\right )^{5/4}} \, dx\\ &=-\frac{8 x}{15 \sqrt [4]{2+3 x^2}}+\frac{2}{15} x \left (2+3 x^2\right )^{3/4}+\frac{8 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{15 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0091322, size = 41, normalized size = 0.65 \[ \frac{2}{15} x \left (\left (3 x^2+2\right )^{3/4}-2^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 31, normalized size = 0.5 \begin{align*}{\frac{2\,x}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{4}}}}-{\frac{2\,{2}^{3/4}x}{15}{\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{x^{2}}{{\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{x^{2}}{{\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 = 0.597907, size = 27, normalized size = 0.43 \begin{align*} \frac{2^{\frac{3}{4}} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{6} \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{x^{2}}{{\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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