Optimal. Leaf size=71 \[ \frac{21}{10} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )-\frac{21 \sqrt{1-x^4}}{10 x}-\frac{7 \sqrt{1-x^4}}{10 x^5}+\frac{1}{2 x^5 \sqrt{1-x^4}}-\frac{21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.0263981, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {290, 325, 307, 221, 1181, 424} \[ -\frac{21 \sqrt{1-x^4}}{10 x}-\frac{7 \sqrt{1-x^4}}{10 x^5}+\frac{1}{2 x^5 \sqrt{1-x^4}}+\frac{21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 307
Rule 221
Rule 1181
Rule 424
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (1-x^4\right )^{3/2}} \, dx &=\frac{1}{2 x^5 \sqrt{1-x^4}}+\frac{7}{2} \int \frac{1}{x^6 \sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^5 \sqrt{1-x^4}}-\frac{7 \sqrt{1-x^4}}{10 x^5}+\frac{21}{10} \int \frac{1}{x^2 \sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^5 \sqrt{1-x^4}}-\frac{7 \sqrt{1-x^4}}{10 x^5}-\frac{21 \sqrt{1-x^4}}{10 x}-\frac{21}{10} \int \frac{x^2}{\sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^5 \sqrt{1-x^4}}-\frac{7 \sqrt{1-x^4}}{10 x^5}-\frac{21 \sqrt{1-x^4}}{10 x}+\frac{21}{10} \int \frac{1}{\sqrt{1-x^4}} \, dx-\frac{21}{10} \int \frac{1+x^2}{\sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^5 \sqrt{1-x^4}}-\frac{7 \sqrt{1-x^4}}{10 x^5}-\frac{21 \sqrt{1-x^4}}{10 x}+\frac{21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{21}{10} \int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx\\ &=\frac{1}{2 x^5 \sqrt{1-x^4}}-\frac{7 \sqrt{1-x^4}}{10 x^5}-\frac{21 \sqrt{1-x^4}}{10 x}-\frac{21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac{21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0029563, size = 20, normalized size = 0.28 \[ -\frac{\, _2F_1\left (-\frac{5}{4},\frac{3}{2};-\frac{1}{4};x^4\right )}{5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 82, normalized size = 1.2 \begin{align*}{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{5\,{x}^{5}}\sqrt{-{x}^{4}+1}}-{\frac{8}{5\,x}\sqrt{-{x}^{4}+1}}+{\frac{21\,{\it EllipticF} \left ( x,i \right ) -21\,{\it EllipticE} \left ( x,i \right ) }{10}\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{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{6}}\,{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^{14} - 2 \, x^{10} + x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.4366, size = 37, normalized size = 0.52 \begin{align*} \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{2} \\ - \frac{1}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{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{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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