Optimal. Leaf size=40 \[ \frac{1}{9} \left (x^6+2\right )^{3/2}-\frac{4 \sqrt{x^6+2}}{3}-\frac{4}{3 \sqrt{x^6+2}} \]
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Rubi [A] time = 0.0152648, antiderivative size = 40, 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} \[ \frac{1}{9} \left (x^6+2\right )^{3/2}-\frac{4 \sqrt{x^6+2}}{3}-\frac{4}{3 \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{17}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{x^2}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{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*}
Mathematica [A] time = 0.0078036, size = 23, normalized size = 0.57 \[ \frac{x^{12}-8 x^6-32}{9 \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 20, normalized size = 0.5 \begin{align*}{\frac{{x}^{12}-8\,{x}^{6}-32}{9}{\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 = 0.976307, size = 38, normalized size = 0.95 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41152, size = 53, normalized size = 1.32 \begin{align*} \frac{x^{12} - 8 \, x^{6} - 32}{9 \, \sqrt{x^{6} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.01411, size = 39, normalized size = 0.98 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14306, size = 38, normalized size = 0.95 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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