Optimal. Leaf size=92 \[ \frac{15 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{28 \sqrt{x^4+1}}+\frac{15 \sqrt{x^4+1}}{14 x^3}-\frac{9 \sqrt{x^4+1}}{14 x^7}+\frac{1}{2 x^7 \sqrt{x^4+1}} \]
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Rubi [A] time = 0.0212835, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 220} \[ \frac{15 \sqrt{x^4+1}}{14 x^3}-\frac{9 \sqrt{x^4+1}}{14 x^7}+\frac{1}{2 x^7 \sqrt{x^4+1}}+\frac{15 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x^8 \left (1+x^4\right )^{3/2}} \, dx &=\frac{1}{2 x^7 \sqrt{1+x^4}}+\frac{9}{2} \int \frac{1}{x^8 \sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x^7 \sqrt{1+x^4}}-\frac{9 \sqrt{1+x^4}}{14 x^7}-\frac{45}{14} \int \frac{1}{x^4 \sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x^7 \sqrt{1+x^4}}-\frac{9 \sqrt{1+x^4}}{14 x^7}+\frac{15 \sqrt{1+x^4}}{14 x^3}+\frac{15}{14} \int \frac{1}{\sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x^7 \sqrt{1+x^4}}-\frac{9 \sqrt{1+x^4}}{14 x^7}+\frac{15 \sqrt{1+x^4}}{14 x^3}+\frac{15 \left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{1+x^4}}\\ \end{align*}
Mathematica [C] time = 0.002711, size = 22, normalized size = 0.24 \[ -\frac{\, _2F_1\left (-\frac{7}{4},\frac{3}{2};-\frac{3}{4};-x^4\right )}{7 x^7} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.047, size = 96, normalized size = 1. \begin{align*} -{\frac{1}{7\,{x}^{7}}\sqrt{{x}^{4}+1}}+{\frac{4}{7\,{x}^{3}}\sqrt{{x}^{4}+1}}+{\frac{x}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{15\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{7\,\sqrt{2}+7\,i\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\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^{8}}\,{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^{16} + 2 \, x^{12} + x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.78596, size = 36, normalized size = 0.39 \begin{align*} \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{3}{2} \\ - \frac{3}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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