Optimal. Leaf size=29 \[ -\frac {1}{3} x \sqrt {16-x^4}+\frac {8}{3} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 29, 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, 227}
\begin {gather*} \frac {8}{3} F\left (\left .\text {ArcSin}\left (\frac {x}{2}\right )\right |-1\right )-\frac {1}{3} x \sqrt {16-x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 327
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {16-x^4}} \, dx &=-\frac {1}{3} x \sqrt {16-x^4}+\frac {16}{3} \int \frac {1}{\sqrt {16-x^4}} \, dx\\ &=-\frac {1}{3} x \sqrt {16-x^4}+\frac {8}{3} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 36, normalized size = 1.24 \begin {gather*} -\frac {1}{3} x \left (\sqrt {16-x^4}-4 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {x^4}{16}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 46 vs. \(2 (21 ) = 42\).
time = 0.16, size = 47, normalized size = 1.62
method | result | size |
meijerg | \(\frac {x^{5} \hypergeom \left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {9}{4}\right ], \frac {x^{4}}{16}\right )}{20}\) | \(17\) |
default | \(-\frac {x \sqrt {-x^{4}+16}}{3}+\frac {8 \sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x}{2}, i\right )}{3 \sqrt {-x^{4}+16}}\) | \(47\) |
elliptic | \(-\frac {x \sqrt {-x^{4}+16}}{3}+\frac {8 \sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x}{2}, i\right )}{3 \sqrt {-x^{4}+16}}\) | \(47\) |
risch | \(\frac {x \left (x^{4}-16\right )}{3 \sqrt {-x^{4}+16}}+\frac {8 \sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x}{2}, i\right )}{3 \sqrt {-x^{4}+16}}\) | \(52\) |
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 [A]
time = 0.08, size = 12, normalized size = 0.41 \begin {gather*} -\frac {1}{3} \, \sqrt {-x^{4} + 16} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 32, normalized size = 1.10 \begin {gather*} \frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac {9}{4}\right )} \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 = -0.03 \begin {gather*} \int \frac {x^4}{\sqrt {16-x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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