Optimal. Leaf size=102 \[ \frac {2\ 2^{3/4} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{9 \sqrt {3} x}+\frac {2}{9} \sqrt [4]{3 x^2-2} x \]
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Rubi [A] time = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {321, 234, 220} \[ \frac {2}{9} \sqrt [4]{3 x^2-2} x+\frac {2\ 2^{3/4} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{9 \sqrt {3} x} \]
Antiderivative was successfully verified.
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Rule 220
Rule 234
Rule 321
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-2+3 x^2\right )^{3/4}} \, dx &=\frac {2}{9} x \sqrt [4]{-2+3 x^2}+\frac {4}{9} \int \frac {1}{\left (-2+3 x^2\right )^{3/4}} \, dx\\ &=\frac {2}{9} x \sqrt [4]{-2+3 x^2}+\frac {\left (4 \sqrt {\frac {2}{3}} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{9 x}\\ &=\frac {2}{9} x \sqrt [4]{-2+3 x^2}+\frac {2\ 2^{3/4} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{9 \sqrt {3} x}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 57, normalized size = 0.56 \[ \frac {2 x \left (\sqrt [4]{2} \left (2-3 x^2\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\frac {3 x^2}{2}\right )+3 x^2-2\right )}{9 \left (3 x^2-2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}}}, x\right ) \]
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 \[ \int \frac {x^{2}}{{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 53, normalized size = 0.52 \[ \frac {2 \,2^{\frac {1}{4}} \left (-\mathrm {signum}\left (\frac {3 x^{2}}{2}-1\right )\right )^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{9 \mathrm {signum}\left (\frac {3 x^{2}}{2}-1\right )^{\frac {3}{4}}}+\frac {2 \left (3 x^{2}-2\right )^{\frac {1}{4}} x}{9} \]
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 \[ \int \frac {x^{2}}{{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\left (3\,x^2-2\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.74, size = 31, normalized size = 0.30 \[ \frac {\sqrt [4]{2} x^{3} e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {3 x^{2}}{2}} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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