Optimal. Leaf size=65 \[ \frac {16\ 2^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right ),2\right )}{63 \sqrt {3}}-\frac {8}{63} \sqrt [4]{3 x^2+2} x+\frac {2}{21} \sqrt [4]{3 x^2+2} x^3 \]
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Rubi [A] time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {321, 231} \[ \frac {2}{21} \sqrt [4]{3 x^2+2} x^3-\frac {8}{63} \sqrt [4]{3 x^2+2} x+\frac {16\ 2^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{63 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 231
Rule 321
Rubi steps
\begin {align*} \int \frac {x^4}{\left (2+3 x^2\right )^{3/4}} \, dx &=\frac {2}{21} x^3 \sqrt [4]{2+3 x^2}-\frac {4}{7} \int \frac {x^2}{\left (2+3 x^2\right )^{3/4}} \, dx\\ &=-\frac {8}{63} x \sqrt [4]{2+3 x^2}+\frac {2}{21} x^3 \sqrt [4]{2+3 x^2}+\frac {16}{63} \int \frac {1}{\left (2+3 x^2\right )^{3/4}} \, dx\\ &=-\frac {8}{63} x \sqrt [4]{2+3 x^2}+\frac {2}{21} x^3 \sqrt [4]{2+3 x^2}+\frac {16\ 2^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{63 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 49, normalized size = 0.75 \[ \frac {2}{63} x \left (4 \sqrt [4]{2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {3 x^2}{2}\right )+\sqrt [4]{3 x^2+2} \left (3 x^2-4\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}}, x\right ) \]
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 \[ \int \frac {x^{4}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 38, normalized size = 0.58 \[ \frac {8 \,2^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{63}+\frac {2 \left (3 x^{2}-4\right ) \left (3 x^{2}+2\right )^{\frac {1}{4}} x}{63} \]
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 \[ \int \frac {x^{4}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {x^4}{{\left (3\,x^2+2\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.74, size = 27, normalized size = 0.42 \[ \frac {\sqrt [4]{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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