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 = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 215
Rule 275
Rule 321
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*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ \frac {1}{6} x^3 \sqrt {x^6+2}-\frac {1}{3} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 29, normalized size = 0.94 \[ \frac {1}{6} \, \sqrt {x^{6} + 2} x^{3} + \frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{8}}{\sqrt {x^{6} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 25, normalized size = 0.81 \[ \frac {\sqrt {x^{6}+2}\, x^{3}}{6}-\frac {\arcsinh \left (\frac {\sqrt {2}\, x^{3}}{2}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.99, size = 58, normalized size = 1.87 \[ \frac {\sqrt {x^{6} + 2}}{3 \, x^{3} {\left (\frac {x^{6} + 2}{x^{6}} - 1\right )}} - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 2}}{x^{3}} + 1\right ) + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 2}}{x^{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^8}{\sqrt {x^6+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.32, size = 39, normalized size = 1.26 \[ \frac {x^{9}}{6 \sqrt {x^{6} + 2}} + \frac {x^{3}}{3 \sqrt {x^{6} + 2}} - \frac {\operatorname {asinh}{\left (\frac {\sqrt {2} x^{3}}{2} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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