Optimal. Leaf size=49 \[ -\frac {1}{18 x^9 \sqrt {2+x^6}}+\frac {1}{9 x^3 \sqrt {2+x^6}}+\frac {x^3}{9 \sqrt {2+x^6}} \]
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Rubi [A]
time = 0.01, antiderivative size = 49, 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 = {277, 270}
\begin {gather*} -\frac {1}{18 \sqrt {x^6+2} x^9}+\frac {x^3}{9 \sqrt {x^6+2}}+\frac {1}{9 \sqrt {x^6+2} x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx &=-\frac {1}{18 x^9 \sqrt {2+x^6}}-\frac {2}{3} \int \frac {1}{x^4 \left (2+x^6\right )^{3/2}} \, dx\\ &=-\frac {1}{18 x^9 \sqrt {2+x^6}}+\frac {1}{9 x^3 \sqrt {2+x^6}}+\frac {2}{3} \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx\\ &=-\frac {1}{18 x^9 \sqrt {2+x^6}}+\frac {1}{9 x^3 \sqrt {2+x^6}}+\frac {x^3}{9 \sqrt {2+x^6}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 28, normalized size = 0.57 \begin {gather*} \frac {-1+2 x^6+2 x^{12}}{18 x^9 \sqrt {2+x^6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 25, normalized size = 0.51
method | result | size |
gosper | \(\frac {2 x^{12}+2 x^{6}-1}{18 x^{9} \sqrt {x^{6}+2}}\) | \(25\) |
trager | \(\frac {2 x^{12}+2 x^{6}-1}{18 x^{9} \sqrt {x^{6}+2}}\) | \(25\) |
risch | \(\frac {2 x^{12}+2 x^{6}-1}{18 x^{9} \sqrt {x^{6}+2}}\) | \(25\) |
meijerg | \(-\frac {\sqrt {2}\, \left (-2 x^{12}-2 x^{6}+1\right )}{36 x^{9} \sqrt {1+\frac {x^{6}}{2}}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 37, normalized size = 0.76 \begin {gather*} \frac {x^{3}}{24 \, \sqrt {x^{6} + 2}} + \frac {\sqrt {x^{6} + 2}}{12 \, x^{3}} - \frac {{\left (x^{6} + 2\right )}^{\frac {3}{2}}}{72 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 44, normalized size = 0.90 \begin {gather*} \frac {2 \, x^{15} + 4 \, x^{9} + {\left (2 \, x^{12} + 2 \, x^{6} - 1\right )} \sqrt {x^{6} + 2}}{18 \, {\left (x^{15} + 2 \, x^{9}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.74, size = 70, normalized size = 1.43 \begin {gather*} \frac {2 x^{12} \sqrt {1 + \frac {2}{x^{6}}}}{18 x^{12} + 36 x^{6}} + \frac {2 x^{6} \sqrt {1 + \frac {2}{x^{6}}}}{18 x^{12} + 36 x^{6}} - \frac {\sqrt {1 + \frac {2}{x^{6}}}}{18 x^{12} + 36 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.24, size = 28, normalized size = 0.57 \begin {gather*} -\frac {24\,x^6-8\,{\left (x^6+2\right )}^2+36}{72\,x^9\,\sqrt {x^6+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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