Optimal. Leaf size=27 \[ \frac {\sqrt {x^6+2}}{3}+\frac {2}{3 \sqrt {x^6+2}} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {\sqrt {x^6+2}}{3}+\frac {2}{3 \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {x}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \left (-\frac {2}{(2+x)^{3/2}}+\frac {1}{\sqrt {2+x}}\right ) \, dx,x,x^6\right )\\ &=\frac {2}{3 \sqrt {2+x^6}}+\frac {\sqrt {2+x^6}}{3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.67 \[ \frac {x^6+4}{3 \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 14, normalized size = 0.52 \[ \frac {x^{6} + 4}{3 \, \sqrt {x^{6} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 19, normalized size = 0.70 \[ \frac {1}{3} \, \sqrt {x^{6} + 2} + \frac {2}{3 \, \sqrt {x^{6} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 15, normalized size = 0.56 \[ \frac {x^{6}+4}{3 \sqrt {x^{6}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 19, normalized size = 0.70 \[ \frac {1}{3} \, \sqrt {x^{6} + 2} + \frac {2}{3 \, \sqrt {x^{6} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 14, normalized size = 0.52 \[ \frac {x^6+4}{3\,\sqrt {x^6+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.82, size = 24, normalized size = 0.89 \[ \frac {x^{6}}{3 \sqrt {x^{6} + 2}} + \frac {4}{3 \sqrt {x^{6} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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