Optimal. Leaf size=51 \[ \frac {8}{3 \sqrt {2+x^6}}+4 \sqrt {2+x^6}-\frac {2}{3} \left (2+x^6\right )^{3/2}+\frac {1}{15} \left (2+x^6\right )^{5/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 51, 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}{15} \left (x^6+2\right )^{5/2}-\frac {2}{3} \left (x^6+2\right )^{3/2}+4 \sqrt {x^6+2}+\frac {8}{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^{23}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {x^3}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6} \text {Subst}\left (\int \left (-\frac {8}{(2+x)^{3/2}}+\frac {12}{\sqrt {2+x}}-6 \sqrt {2+x}+(2+x)^{3/2}\right ) \, dx,x,x^6\right )\\ &=\frac {8}{3 \sqrt {2+x^6}}+4 \sqrt {2+x^6}-\frac {2}{3} \left (2+x^6\right )^{3/2}+\frac {1}{15} \left (2+x^6\right )^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.55 \begin {gather*} \frac {128+32 x^6-4 x^{12}+x^{18}}{15 \sqrt {2+x^6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 25, normalized size = 0.49
method | result | size |
gosper | \(\frac {x^{18}-4 x^{12}+32 x^{6}+128}{15 \sqrt {x^{6}+2}}\) | \(25\) |
trager | \(\frac {x^{18}-4 x^{12}+32 x^{6}+128}{15 \sqrt {x^{6}+2}}\) | \(25\) |
risch | \(\frac {x^{18}-4 x^{12}+32 x^{6}+128}{15 \sqrt {x^{6}+2}}\) | \(25\) |
meijerg | \(\frac {4 \sqrt {2}\, \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (x^{18}-4 x^{12}+32 x^{6}+128\right )}{40 \sqrt {1+\frac {x^{6}}{2}}}\right )}{3 \sqrt {\pi }}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 37, normalized size = 0.73 \begin {gather*} \frac {1}{15} \, {\left (x^{6} + 2\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} + 4 \, \sqrt {x^{6} + 2} + \frac {8}{3 \, \sqrt {x^{6} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 24, normalized size = 0.47 \begin {gather*} \frac {x^{18} - 4 \, x^{12} + 32 \, x^{6} + 128}{15 \, \sqrt {x^{6} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.27, size = 54, normalized size = 1.06 \begin {gather*} \frac {x^{18}}{15 \sqrt {x^{6} + 2}} - \frac {4 x^{12}}{15 \sqrt {x^{6} + 2}} + \frac {32 x^{6}}{15 \sqrt {x^{6} + 2}} + \frac {128}{15 \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 = 37, normalized size = 0.73 \begin {gather*} \frac {1}{15} \, {\left (x^{6} + 2\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} + 4 \, \sqrt {x^{6} + 2} + \frac {8}{3 \, \sqrt {x^{6} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 32, normalized size = 0.63 \begin {gather*} \frac {{\left (x^6+2\right )}^3-10\,{\left (x^6+2\right )}^2+60\,x^6+160}{15\,\sqrt {x^6+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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