Optimal. Leaf size=62 \[ \frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (x^2\right ),\frac{1}{2}\right )}{6 \sqrt{x^8+1}}+\frac{1}{6} \sqrt{x^8+1} x^2 \]
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Rubi [A] time = 0.0245337, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {275, 195, 220} \[ \frac{1}{6} \sqrt{x^8+1} x^2+\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{6 \sqrt{x^8+1}} \]
Antiderivative was successfully verified.
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Rule 275
Rule 195
Rule 220
Rubi steps
\begin{align*} \int x \sqrt{1+x^8} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{1+x^4} \, dx,x,x^2\right )\\ &=\frac{1}{6} x^2 \sqrt{1+x^8}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,x^2\right )\\ &=\frac{1}{6} x^2 \sqrt{1+x^8}+\frac{\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 )}{6 \sqrt{1+x^8}}\\ \end{align*}
Mathematica [C] time = 0.0026061, size = 22, normalized size = 0.35 \[ \frac{1}{2} x^2 \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-x^8\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 30, normalized size = 0.5 \begin{align*}{\frac{{x}^{2}}{6}\sqrt{{x}^{8}+1}}+{\frac{{x}^{2}}{3}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{5}{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 \sqrt{x^{8} + 1} x\,{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 (\sqrt{x^{8} + 1} x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.530719, size = 31, normalized size = 0.5 \begin{align*} \frac{x^{2} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{5}{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 \sqrt{x^{8} + 1} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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