Optimal. Leaf size=40 \[ -\frac {4}{3 \sqrt {2+x^6}}-\frac {4 \sqrt {2+x^6}}{3}+\frac {1}{9} \left (2+x^6\right )^{3/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 40, 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 {1}{9} \left (x^6+2\right )^{3/2}-\frac {4 \sqrt {x^6+2}}{3}-\frac {4}{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^{17}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {x^2}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6} \text {Subst}\left (\int \left (\frac {4}{(2+x)^{3/2}}-\frac {4}{\sqrt {2+x}}+\sqrt {2+x}\right ) \, dx,x,x^6\right )\\ &=-\frac {4}{3 \sqrt {2+x^6}}-\frac {4 \sqrt {2+x^6}}{3}+\frac {1}{9} \left (2+x^6\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.58 \begin {gather*} \frac {-32-8 x^6+x^{12}}{9 \sqrt {2+x^6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 20, normalized size = 0.50
method | result | size |
gosper | \(\frac {x^{12}-8 x^{6}-32}{9 \sqrt {x^{6}+2}}\) | \(20\) |
trager | \(\frac {x^{12}-8 x^{6}-32}{9 \sqrt {x^{6}+2}}\) | \(20\) |
risch | \(\frac {x^{12}-8 x^{6}-32}{9 \sqrt {x^{6}+2}}\) | \(20\) |
meijerg | \(\frac {2 \sqrt {2}\, \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {1}{2} x^{12}+4 x^{6}+16\right )}{6 \sqrt {1+\frac {x^{6}}{2}}}\right )}{3 \sqrt {\pi }}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 28, normalized size = 0.70 \begin {gather*} \frac {1}{9} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - \frac {4}{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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Fricas [A]
time = 0.37, size = 19, normalized size = 0.48 \begin {gather*} \frac {x^{12} - 8 \, x^{6} - 32}{9 \, \sqrt {x^{6} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.66, size = 39, normalized size = 0.98 \begin {gather*} \frac {x^{12}}{9 \sqrt {x^{6} + 2}} - \frac {8 x^{6}}{9 \sqrt {x^{6} + 2}} - \frac {32}{9 \sqrt {x^{6} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 28, normalized size = 0.70 \begin {gather*} \frac {1}{9} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - \frac {4}{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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Mupad [B]
time = 1.18, size = 25, normalized size = 0.62 \begin {gather*} -\frac {12\,x^6-{\left (x^6+2\right )}^2+36}{9\,\sqrt {x^6+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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