Optimal. Leaf size=43 \[ -\frac{96}{5} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-1\right )-\frac{1}{5} \sqrt{16-x^4} x^3+\frac{96}{5} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]
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Rubi [A] time = 0.0201656, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {321, 307, 221, 1181, 21, 424} \[ -\frac{1}{5} \sqrt{16-x^4} x^3-\frac{96}{5} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )+\frac{96}{5} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 321
Rule 307
Rule 221
Rule 1181
Rule 21
Rule 424
Rubi steps
\begin{align*} \int \frac{x^6}{\sqrt{16-x^4}} \, dx &=-\frac{1}{5} x^3 \sqrt{16-x^4}+\frac{48}{5} \int \frac{x^2}{\sqrt{16-x^4}} \, dx\\ &=-\frac{1}{5} x^3 \sqrt{16-x^4}-\frac{192}{5} \int \frac{1}{\sqrt{16-x^4}} \, dx+\frac{192}{5} \int \frac{1+\frac{x^2}{4}}{\sqrt{16-x^4}} \, dx\\ &=-\frac{1}{5} x^3 \sqrt{16-x^4}-\frac{96}{5} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )+\frac{192}{5} \int \frac{1+\frac{x^2}{4}}{\sqrt{4-x^2} \sqrt{4+x^2}} \, dx\\ &=-\frac{1}{5} x^3 \sqrt{16-x^4}-\frac{96}{5} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )+\frac{48}{5} \int \frac{\sqrt{4+x^2}}{\sqrt{4-x^2}} \, dx\\ &=-\frac{1}{5} x^3 \sqrt{16-x^4}+\frac{96}{5} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-\frac{96}{5} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0076516, size = 38, normalized size = 0.88 \[ -\frac{1}{5} x^3 \left (\sqrt{16-x^4}-4 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{x^4}{16}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 58, normalized size = 1.4 \begin{align*} -{\frac{{x}^{3}}{5}\sqrt{-{x}^{4}+16}}-{\frac{96}{5}\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4} \left ({\it EllipticF} \left ({\frac{x}{2}},i \right ) -{\it EllipticE} \left ({\frac{x}{2}},i \right ) \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^{6}}{\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^{6}}{x^{4} - 16}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.930464, size = 32, normalized size = 0.74 \begin{align*} \frac{x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac{11}{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^{6}}{\sqrt{-x^{4} + 16}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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