Optimal. Leaf size=45 \[ -\frac{x^9}{3 \sqrt{x^6+2}}+\frac{1}{2} \sqrt{x^6+2} x^3-\sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0188832, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {275, 288, 321, 215} \[ -\frac{x^9}{3 \sqrt{x^6+2}}+\frac{1}{2} \sqrt{x^6+2} x^3-\sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 288
Rule 321
Rule 215
Rubi steps
\begin{align*} \int \frac{x^{14}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4}{\left (2+x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=-\frac{x^9}{3 \sqrt{2+x^6}}+\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+x^2}} \, dx,x,x^3\right )\\ &=-\frac{x^9}{3 \sqrt{2+x^6}}+\frac{1}{2} x^3 \sqrt{2+x^6}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{2+x^2}} \, dx,x,x^3\right )\\ &=-\frac{x^9}{3 \sqrt{2+x^6}}+\frac{1}{2} x^3 \sqrt{2+x^6}-\sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.009023, size = 43, normalized size = 0.96 \[ \frac{x^9+6 x^3-6 \sqrt{x^6+2} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )}{6 \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 30, normalized size = 0.7 \begin{align*}{\frac{{x}^{3} \left ({x}^{6}+6 \right ) }{6}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\it Arcsinh} \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.988742, size = 99, normalized size = 2.2 \begin{align*} -\frac{\frac{3 \,{\left (x^{6} + 2\right )}}{x^{6}} - 2}{3 \,{\left (\frac{\sqrt{x^{6} + 2}}{x^{3}} - \frac{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}{x^{9}}\right )}} - \frac{1}{2} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} + 1\right ) + \frac{1}{2} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51649, size = 131, normalized size = 2.91 \begin{align*} \frac{4 \, x^{6} + 6 \,{\left (x^{6} + 2\right )} \log \left (-x^{3} + \sqrt{x^{6} + 2}\right ) +{\left (x^{9} + 6 \, x^{3}\right )} \sqrt{x^{6} + 2} + 8}{6 \,{\left (x^{6} + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.6999, size = 36, normalized size = 0.8 \begin{align*} \frac{x^{9}}{6 \sqrt{x^{6} + 2}} + \frac{x^{3}}{\sqrt{x^{6} + 2}} - \operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{14}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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