Optimal. Leaf size=394 \[ \frac{5 \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right ),-7-4 \sqrt{3}\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}+\frac{5 \sqrt{x^6+2}}{24 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{5 \sqrt{x^6+2}}{24 x^2}+\frac{1}{6 x^2 \sqrt{x^6+2}}-\frac{5 \sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{8\ 2^{5/6} 3^{3/4} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]
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Rubi [A] time = 0.220999, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {275, 290, 325, 303, 218, 1877} \[ \frac{5 \sqrt{x^6+2}}{24 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{5 \sqrt{x^6+2}}{24 x^2}+\frac{1}{6 x^2 \sqrt{x^6+2}}+\frac{5 \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}-\frac{5 \sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{8\ 2^{5/6} 3^{3/4} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
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Rule 275
Rule 290
Rule 325
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (2+x^3\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{6 x^2 \sqrt{2+x^6}}+\frac{5}{12} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{2+x^3}} \, dx,x,x^2\right )\\ &=\frac{1}{6 x^2 \sqrt{2+x^6}}-\frac{5 \sqrt{2+x^6}}{24 x^2}+\frac{5}{48} \operatorname{Subst}\left (\int \frac{x}{\sqrt{2+x^3}} \, dx,x,x^2\right )\\ &=\frac{1}{6 x^2 \sqrt{2+x^6}}-\frac{5 \sqrt{2+x^6}}{24 x^2}+\frac{5}{48} \operatorname{Subst}\left (\int \frac{\sqrt [3]{2} \left (1-\sqrt{3}\right )+x}{\sqrt{2+x^3}} \, dx,x,x^2\right )+\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+x^3}} \, dx,x,x^2\right )}{24 \sqrt [6]{2} \sqrt{2+\sqrt{3}}}\\ &=\frac{1}{6 x^2 \sqrt{2+x^6}}-\frac{5 \sqrt{2+x^6}}{24 x^2}+\frac{5 \sqrt{2+x^6}}{24 \left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )}-\frac{5 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{2}+x^2\right ) \sqrt{\frac{2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{2} \left (1-\sqrt{3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2}\right )|-7-4 \sqrt{3}\right )}{8\ 2^{5/6} 3^{3/4} \sqrt{\frac{\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )^2}} \sqrt{2+x^6}}+\frac{5 \left (\sqrt [3]{2}+x^2\right ) \sqrt{\frac{2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{2} \left (1-\sqrt{3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2}\right )|-7-4 \sqrt{3}\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )^2}} \sqrt{2+x^6}}\\ \end{align*}
Mathematica [C] time = 0.0048871, size = 29, normalized size = 0.07 \[ -\frac{\, _2F_1\left (-\frac{1}{3},\frac{3}{2};\frac{2}{3};-\frac{x^6}{2}\right )}{4 \sqrt{2} x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 40, normalized size = 0.1 \begin{align*} -{\frac{5\,{x}^{6}+6}{24\,{x}^{2}}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{5\,{x}^{4}\sqrt{2}}{192}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{2}{3}};\,{\frac{5}{3}};\,-{\frac{{x}^{6}}{2}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{6} + 2}}{x^{15} + 4 \, x^{9} + 4 \, x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.777815, size = 39, normalized size = 0.1 \begin{align*} \frac{\sqrt{2} \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{3}{2} \\ \frac{2}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 x^{2} \Gamma \left (\frac{2}{3}\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{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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