Optimal. Leaf size=31 \[ \frac{1}{6} x^3 \sqrt{x^6+2}-\frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0116619, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 321, 215} \[ \frac{1}{6} x^3 \sqrt{x^6+2}-\frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 215
Rubi steps
\begin{align*} \int \frac{x^8}{\sqrt{2+x^6}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+x^2}} \, dx,x,x^3\right )\\ &=\frac{1}{6} x^3 \sqrt{2+x^6}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+x^2}} \, dx,x,x^3\right )\\ &=\frac{1}{6} x^3 \sqrt{2+x^6}-\frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0049871, size = 31, normalized size = 1. \[ \frac{1}{6} x^3 \sqrt{x^6+2}-\frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 25, normalized size = 0.8 \begin{align*} -{\frac{1}{3}{\it Arcsinh} \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ) }+{\frac{{x}^{3}}{6}\sqrt{{x}^{6}+2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01733, size = 78, normalized size = 2.52 \begin{align*} \frac{\sqrt{x^{6} + 2}}{3 \, x^{3}{\left (\frac{x^{6} + 2}{x^{6}} - 1\right )}} - \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} + 1\right ) + \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47324, size = 74, normalized size = 2.39 \begin{align*} \frac{1}{6} \, \sqrt{x^{6} + 2} x^{3} + \frac{1}{3} \, \log \left (-x^{3} + \sqrt{x^{6} + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.89741, size = 39, normalized size = 1.26 \begin{align*} \frac{x^{9}}{6 \sqrt{x^{6} + 2}} + \frac{x^{3}}{3 \sqrt{x^{6} + 2}} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} \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}}{\sqrt{x^{6} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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