Optimal. Leaf size=130 \[ -\frac{3 \left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (x^2\right ),\frac{1}{2}\right )}{20 \sqrt{x^8+1}}+\frac{1}{10} \sqrt{x^8+1} x^6-\frac{3 \sqrt{x^8+1} x^2}{10 \left (x^4+1\right )}+\frac{3 \left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{10 \sqrt{x^8+1}} \]
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Rubi [A] time = 0.0529887, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {275, 321, 305, 220, 1196} \[ \frac{1}{10} \sqrt{x^8+1} x^6-\frac{3 \sqrt{x^8+1} x^2}{10 \left (x^4+1\right )}-\frac{3 \left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{20 \sqrt{x^8+1}}+\frac{3 \left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{10 \sqrt{x^8+1}} \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^{13}}{\sqrt{1+x^8}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{1+x^4}} \, dx,x,x^2\right )\\ &=\frac{1}{10} x^6 \sqrt{1+x^8}-\frac{3}{10} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^4}} \, dx,x,x^2\right )\\ &=\frac{1}{10} x^6 \sqrt{1+x^8}-\frac{3}{10} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,x^2\right )+\frac{3}{10} \operatorname{Subst}\left (\int \frac{1-x^2}{\sqrt{1+x^4}} \, dx,x,x^2\right )\\ &=\frac{1}{10} x^6 \sqrt{1+x^8}-\frac{3 x^2 \sqrt{1+x^8}}{10 \left (1+x^4\right )}+\frac{3 \left (1+x^4\right ) \sqrt{\frac{1+x^8}{\left (1+x^4\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{10 \sqrt{1+x^8}}-\frac{3 \left (1+x^4\right ) \sqrt{\frac{1+x^8}{\left (1+x^4\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{20 \sqrt{1+x^8}}\\ \end{align*}
Mathematica [C] time = 0.006112, size = 34, normalized size = 0.26 \[ \frac{1}{10} x^6 \left (\sqrt{x^8+1}-\, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-x^8\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 30, normalized size = 0.2 \begin{align*}{\frac{{x}^{6}}{10}\sqrt{{x}^{8}+1}}-{\frac{{x}^{6}}{10}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{7}{4}};\,-{x}^{8})}} \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^{13}}{\sqrt{x^{8} + 1}}\,{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{x^{13}}{\sqrt{x^{8} + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.40793, size = 29, normalized size = 0.22 \begin{align*} \frac{x^{14} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \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^{13}}{\sqrt{x^{8} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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