Optimal. Leaf size=65 \[ -\frac {8}{135} \left (2-3 x^2\right )^{3/4} x-\frac {2}{27} \left (2-3 x^2\right )^{3/4} x^3+\frac {32 \sqrt [4]{2} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{135 \sqrt {3}} \]
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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, 228} \[ -\frac {2}{27} \left (2-3 x^2\right )^{3/4} x^3-\frac {8}{135} \left (2-3 x^2\right )^{3/4} x+\frac {32 \sqrt [4]{2} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{135 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 228
Rule 321
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{2-3 x^2}} \, dx &=-\frac {2}{27} x^3 \left (2-3 x^2\right )^{3/4}+\frac {4}{9} \int \frac {x^2}{\sqrt [4]{2-3 x^2}} \, dx\\ &=-\frac {8}{135} x \left (2-3 x^2\right )^{3/4}-\frac {2}{27} x^3 \left (2-3 x^2\right )^{3/4}+\frac {16}{135} \int \frac {1}{\sqrt [4]{2-3 x^2}} \, dx\\ &=-\frac {8}{135} x \left (2-3 x^2\right )^{3/4}-\frac {2}{27} x^3 \left (2-3 x^2\right )^{3/4}+\frac {32 \sqrt [4]{2} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{135 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 49, normalized size = 0.75 \[ -\frac {2}{135} x \left (\left (2-3 x^2\right )^{3/4} \left (5 x^2+4\right )-4\ 2^{3/4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {3 x^2}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{4}}{3 \, x^{2} - 2}, x\right ) \]
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 \[ \int \frac {x^{4}}{{\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 45, normalized size = 0.69 \[ \frac {8 \,2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{135}+\frac {2 \left (5 x^{2}+4\right ) \left (3 x^{2}-2\right ) x}{135 \left (-3 x^{2}+2\right )^{\frac {1}{4}}} \]
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 \[ \int \frac {x^{4}}{{\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \]
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 \[ \int \frac {x^4}{{\left (2-3\,x^2\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.77, size = 29, normalized size = 0.45 \[ \frac {2^{\frac {3}{4}} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{2 i \pi }}{2}} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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