Optimal. Leaf size=29 \[ \frac{8}{3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-1\right )-\frac{1}{3} x \sqrt{16-x^4} \]
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Rubi [A] time = 0.0057608, antiderivative size = 29, normalized size of antiderivative = 1., 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 321
Rule 221
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*}
Mathematica [C] time = 0.006046, 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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Maple [B] time = 0.007, size = 47, normalized size = 1.6 \begin{align*} -{\frac{x}{3}\sqrt{-{x}^{4}+16}}+{\frac{8}{3}\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4}{\it EllipticF} \left ({\frac{x}{2}},i \right ){\frac{1}{\sqrt{-{x}^{4}+16}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{-x^{4} + 16}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 16} x^{4}}{x^{4} - 16}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.82483, size = 32, normalized size = 1.1 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{-x^{4} + 16}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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