Optimal. Leaf size=43 \[ -\frac{5}{6} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )+\frac{x^5}{2 \sqrt{1-x^4}}+\frac{5}{6} \sqrt{1-x^4} x \]
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Rubi [A] time = 0.0095008, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {288, 321, 221} \[ \frac{x^5}{2 \sqrt{1-x^4}}+\frac{5}{6} \sqrt{1-x^4} x-\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 288
Rule 321
Rule 221
Rubi steps
\begin{align*} \int \frac{x^8}{\left (1-x^4\right )^{3/2}} \, dx &=\frac{x^5}{2 \sqrt{1-x^4}}-\frac{5}{2} \int \frac{x^4}{\sqrt{1-x^4}} \, dx\\ &=\frac{x^5}{2 \sqrt{1-x^4}}+\frac{5}{6} x \sqrt{1-x^4}-\frac{5}{6} \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=\frac{x^5}{2 \sqrt{1-x^4}}+\frac{5}{6} x \sqrt{1-x^4}-\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0107133, size = 49, normalized size = 1.14 \[ -\frac{x \left (5 \sqrt{1-x^4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right )+2 x^4-5\right )}{6 \sqrt{1-x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 57, normalized size = 1.3 \begin{align*}{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{x}{3}\sqrt{-{x}^{4}+1}}-{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{6}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \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^{8}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{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} + 1} x^{8}}{x^{8} - 2 \, x^{4} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.951009, size = 31, normalized size = 0.72 \begin{align*} \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{13}{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^{8}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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