Optimal. Leaf size=47 \[ \frac {2}{9} x \sqrt [4]{3 x^2+2}-\frac {4\ 2^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right ),2\right )}{9 \sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 47, 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 = {321, 231} \[ \frac {2}{9} x \sqrt [4]{3 x^2+2}-\frac {4\ 2^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 231
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 {4\ 2^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 41, normalized size = 0.87 \[ \frac {2}{9} x \left (\sqrt [4]{3 x^2+2}-\sqrt [4]{2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {3 x^2}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, 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 = 31, normalized size = 0.66 \[ -\frac {2 \,2^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{9}+\frac {2 \left (3 x^{2}+2\right )^{\frac {1}{4}} x}{9} \]
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 = -0.02 \[ \int \frac {x^2}{{\left (3\,x^2+2\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.71, size = 27, normalized size = 0.57 \[ \frac {\sqrt [4]{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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