Optimal. Leaf size=47 \[ -\frac{\left (2-3 x^2\right )^{3/4}}{2 x}-\frac{\sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2^{3/4}} \]
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Rubi [A] time = 0.0082851, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {325, 228} \[ -\frac{\left (2-3 x^2\right )^{3/4}}{2 x}-\frac{\sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2^{3/4}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 228
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{\left (2-3 x^2\right )^{3/4}}{2 x}-\frac{\sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0043975, size = 27, normalized size = 0.57 \[ -\frac{\, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{3 x^2}{2}\right )}{\sqrt [4]{2} x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 40, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{2}-2}{2\,x}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}-{\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 = 0.677284, size = 31, normalized size = 0.66 \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^{2 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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