Optimal. Leaf size=51 \[ \frac {1}{15} \left (x^6+2\right )^{5/2}-\frac {2}{3} \left (x^6+2\right )^{3/2}+4 \sqrt {x^6+2}+\frac {8}{3 \sqrt {x^6+2}} \]
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Rubi [A] time = 0.02, 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 = {266, 43} \[ \frac {1}{15} \left (x^6+2\right )^{5/2}-\frac {2}{3} \left (x^6+2\right )^{3/2}+4 \sqrt {x^6+2}+\frac {8}{3 \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{23}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {x^3}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6} \operatorname {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 = 0.01, size = 28, normalized size = 0.55 \[ \frac {x^{18}-4 x^{12}+32 x^6+128}{15 \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 24, normalized size = 0.47 \[ \frac {x^{18} - 4 \, x^{12} + 32 \, x^{6} + 128}{15 \, \sqrt {x^{6} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 37, normalized size = 0.73 \[ \frac {1}{15} \, {\left (x^{6} + 2\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} + 4 \, \sqrt {x^{6} + 2} + \frac {8}{3 \, \sqrt {x^{6} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.49 \[ \frac {x^{18}-4 x^{12}+32 x^{6}+128}{15 \sqrt {x^{6}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 37, normalized size = 0.73 \[ \frac {1}{15} \, {\left (x^{6} + 2\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} + 4 \, \sqrt {x^{6} + 2} + \frac {8}{3 \, \sqrt {x^{6} + 2}} \]
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 \[ \frac {{\left (x^6+2\right )}^3-10\,{\left (x^6+2\right )}^2+60\,x^6+160}{15\,\sqrt {x^6+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.15, size = 54, normalized size = 1.06 \[ \frac {x^{18}}{15 \sqrt {x^{6} + 2}} - \frac {4 x^{12}}{15 \sqrt {x^{6} + 2}} + \frac {32 x^{6}}{15 \sqrt {x^{6} + 2}} + \frac {128}{15 \sqrt {x^{6} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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