Optimal. Leaf size=83 \[ -\frac{9 x}{8 \sqrt [4]{3 x^2+2}}+\frac{3 \left (3 x^2+2\right )^{3/4}}{8 x}-\frac{\left (3 x^2+2\right )^{3/4}}{6 x^3}+\frac{3 \sqrt{3} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{4\ 2^{3/4}} \]
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Rubi [A] time = 0.0192217, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {325, 227, 196} \[ -\frac{9 x}{8 \sqrt [4]{3 x^2+2}}+\frac{3 \left (3 x^2+2\right )^{3/4}}{8 x}-\frac{\left (3 x^2+2\right )^{3/4}}{6 x^3}+\frac{3 \sqrt{3} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{4\ 2^{3/4}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt [4]{2+3 x^2}} \, dx &=-\frac{\left (2+3 x^2\right )^{3/4}}{6 x^3}-\frac{3}{4} \int \frac{1}{x^2 \sqrt [4]{2+3 x^2}} \, dx\\ &=-\frac{\left (2+3 x^2\right )^{3/4}}{6 x^3}+\frac{3 \left (2+3 x^2\right )^{3/4}}{8 x}-\frac{9}{16} \int \frac{1}{\sqrt [4]{2+3 x^2}} \, dx\\ &=-\frac{9 x}{8 \sqrt [4]{2+3 x^2}}-\frac{\left (2+3 x^2\right )^{3/4}}{6 x^3}+\frac{3 \left (2+3 x^2\right )^{3/4}}{8 x}+\frac{9}{8} \int \frac{1}{\left (2+3 x^2\right )^{5/4}} \, dx\\ &=-\frac{9 x}{8 \sqrt [4]{2+3 x^2}}-\frac{\left (2+3 x^2\right )^{3/4}}{6 x^3}+\frac{3 \left (2+3 x^2\right )^{3/4}}{8 x}+\frac{3 \sqrt{3} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{4\ 2^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.006043, size = 29, normalized size = 0.35 \[ -\frac{\, _2F_1\left (-\frac{3}{2},\frac{1}{4};-\frac{1}{2};-\frac{3 x^2}{2}\right )}{3 \sqrt [4]{2} x^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 45, normalized size = 0.5 \begin{align*}{\frac{27\,{x}^{4}+6\,{x}^{2}-8}{24\,{x}^{3}}{\frac{1}{\sqrt [4]{3\,{x}^{2}+2}}}}-{\frac{9\,{2}^{3/4}x}{32}{\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^{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{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}{3 \, x^{6} + 2 \, x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.814624, size = 32, normalized size = 0.39 \begin{align*} - \frac{2^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{6 x^{3}} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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