Optimal. Leaf size=47 \[ -\frac {2}{15} x \left (2-3 x^2\right )^{3/4}+\frac {8 \sqrt [4]{2} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{15 \sqrt {3}} \]
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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 = {327, 234}
\begin {gather*} \frac {8 \sqrt [4]{2} E\left (\left .\frac {1}{2} \text {ArcSin}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{15 \sqrt {3}}-\frac {2}{15} x \left (2-3 x^2\right )^{3/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 234
Rule 327
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt [4]{2-3 x^2}} \, dx &=-\frac {2}{15} x \left (2-3 x^2\right )^{3/4}+\frac {4}{15} \int \frac {1}{\sqrt [4]{2-3 x^2}} \, dx\\ &=-\frac {2}{15} x \left (2-3 x^2\right )^{3/4}+\frac {8 \sqrt [4]{2} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{15 \sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 5.75, size = 41, normalized size = 0.87 \begin {gather*} -\frac {2}{15} x \left (\left (2-3 x^2\right )^{3/4}-2^{3/4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {3 x^2}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.07, size = 20, normalized size = 0.43
method | result | size |
meijerg | \(\frac {2^{\frac {3}{4}} x^{3} \hypergeom \left (\left [\frac {1}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], \frac {3 x^{2}}{2}\right )}{6}\) | \(20\) |
risch | \(\frac {2 x \left (3 x^{2}-2\right )}{15 \left (-3 x^{2}+2\right )^{\frac {1}{4}}}+\frac {2 \,2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{15}\) | \(38\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.44, size = 29, normalized size = 0.62 \begin {gather*} \frac {2^{\frac {3}{4}} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{2 i \pi }}{2}} \right )}}{6} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2}{{\left (2-3\,x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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