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 = 0.0183523, antiderivative size = 51, normalized size of antiderivative = 1., 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 266
Rule 43
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*}
Mathematica [A] time = 0.009383, 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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Maple [A] time = 0.006, size = 25, normalized size = 0.5 \begin{align*}{\frac{{x}^{18}-4\,{x}^{12}+32\,{x}^{6}+128}{15}{\frac{1}{\sqrt{{x}^{6}+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98113, size = 50, normalized size = 0.98 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43175, size = 69, normalized size = 1.35 \begin{align*} \frac{x^{18} - 4 \, x^{12} + 32 \, x^{6} + 128}{15 \, \sqrt{x^{6} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.7386, size = 54, normalized size = 1.06 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1653, size = 50, normalized size = 0.98 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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