Optimal. Leaf size=29 \[ \frac {8}{3} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )-\frac {1}{3} x \sqrt {16-x^4} \]
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Rubi [A] time = 0.01, 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 = {321, 221} \[ \frac {8}{3} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )-\frac {1}{3} x \sqrt {16-x^4} \]
Antiderivative was successfully verified.
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Rule 221
Rule 321
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] time = 0.01, size = 36, normalized size = 1.24 \[ -\frac {1}{3} x \left (\sqrt {16-x^4}-4 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {x^4}{16}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 16} x^{4}}{x^{4} - 16}, x\right ) \]
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 \[ \int \frac {x^{4}}{\sqrt {-x^{4} + 16}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 47, normalized size = 1.62 \[ -\frac {\sqrt {-x^{4}+16}\, x}{3}+\frac {8 \sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x}{2}, i\right )}{3 \sqrt {-x^{4}+16}} \]
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 \[ \int \frac {x^{4}}{\sqrt {-x^{4} + 16}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^4}{\sqrt {16-x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 32, normalized size = 1.10 \[ \frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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