Optimal. Leaf size=49 \[ \frac{\sqrt{3} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{2 \sqrt [4]{2}}-\frac{\sqrt [4]{2-3 x^2}}{2 x} \]
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Rubi [A] time = 0.0089599, antiderivative size = 49, 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, 232} \[ \frac{\sqrt{3} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}}-\frac{\sqrt [4]{2-3 x^2}}{2 x} \]
Antiderivative was successfully verified.
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Rule 325
Rule 232
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (2-3 x^2\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{2-3 x^2}}{2 x}+\frac{3}{4} \int \frac{1}{\left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{2-3 x^2}}{2 x}+\frac{\sqrt{3} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}}\\ \end{align*}
Mathematica [C] time = 0.0049404, size = 27, normalized size = 0.55 \[ -\frac{\, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{1}{2};\frac{3 x^2}{2}\right )}{2^{3/4} x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.014, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \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{3}{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{1}{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 = 0.768468, size = 31, normalized size = 0.63 \begin{align*} - \frac{\sqrt [4]{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{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{3}{4}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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