Optimal. Leaf size=49 \[ -\frac {\sqrt {3} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right ),2\right )}{2 \sqrt [4]{2}}-\frac {\sqrt [4]{3 x^2+2}}{2 x} \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, 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, 231} \[ -\frac {\sqrt [4]{3 x^2+2}}{2 x}-\frac {\sqrt {3} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
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Rule 231
Rule 325
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} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [C] time = 0.01, 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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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{3 \, x^{4} + 2 \, x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 33, normalized size = 0.67 \[ -\frac {3 \,2^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{8}-\frac {\left (3 x^{2}+2\right )^{\frac {1}{4}}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.99, size = 36, normalized size = 0.73 \[ -\frac {2\,3^{1/4}\,{\left (\frac {2}{x^2}+3\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{4},\frac {5}{4};\ \frac {9}{4};\ -\frac {2}{3\,x^2}\right )}{15\,x\,{\left (3\,x^2+2\right )}^{3/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.82, size = 29, normalized size = 0.59 \[ - \frac {\sqrt [4]{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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