Optimal. Leaf size=27 \[ \frac {2}{3 \sqrt {2+x^6}}+\frac {\sqrt {2+x^6}}{3} \]
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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 = {272, 45}
\begin {gather*} \frac {\sqrt {x^6+2}}{3}+\frac {2}{3 \sqrt {x^6+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {x}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6} \text {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 = 0.02, size = 18, normalized size = 0.67 \begin {gather*} \frac {4+x^6}{3 \sqrt {2+x^6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 15, normalized size = 0.56
method | result | size |
gosper | \(\frac {x^{6}+4}{3 \sqrt {x^{6}+2}}\) | \(15\) |
trager | \(\frac {x^{6}+4}{3 \sqrt {x^{6}+2}}\) | \(15\) |
risch | \(\frac {x^{6}+4}{3 \sqrt {x^{6}+2}}\) | \(15\) |
meijerg | \(\frac {\sqrt {2}\, \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (2 x^{6}+8\right )}{4 \sqrt {1+\frac {x^{6}}{2}}}\right )}{3 \sqrt {\pi }}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} + 2} + \frac {2}{3 \, \sqrt {x^{6} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 14, normalized size = 0.52 \begin {gather*} \frac {x^{6} + 4}{3 \, \sqrt {x^{6} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{6}}{3 \sqrt {x^{6} + 2}} + \frac {4}{3 \sqrt {x^{6} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.25, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} + 2} + \frac {2}{3 \, \sqrt {x^{6} + 2}} \end {gather*}
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 \begin {gather*} \frac {x^6+4}{3\,\sqrt {x^6+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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