Optimal. Leaf size=197 \[ -\frac{2^{2/3} x \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \text{EllipticF}\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{15 \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}-\frac{2 \sqrt{x^6+2}}{15 x^5}+\frac{1}{6 x^5 \sqrt{x^6+2}} \]
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Rubi [A] time = 0.04583, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 225} \[ -\frac{2 \sqrt{x^6+2}}{15 x^5}+\frac{1}{6 x^5 \sqrt{x^6+2}}-\frac{2^{2/3} x \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{15 \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 225
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{6 x^5 \sqrt{2+x^6}}+\frac{4}{3} \int \frac{1}{x^6 \sqrt{2+x^6}} \, dx\\ &=\frac{1}{6 x^5 \sqrt{2+x^6}}-\frac{2 \sqrt{2+x^6}}{15 x^5}-\frac{4}{15} \int \frac{1}{\sqrt{2+x^6}} \, dx\\ &=\frac{1}{6 x^5 \sqrt{2+x^6}}-\frac{2 \sqrt{2+x^6}}{15 x^5}-\frac{2^{2/3} x \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 (\cos ^{-1}\left (\frac{\sqrt [3]{2}+\left (1-\sqrt{3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{15 \sqrt [4]{3} \sqrt{\frac{x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2\right )^2}} \sqrt{2+x^6}}\\ \end{align*}
Mathematica [C] time = 0.0044873, size = 29, normalized size = 0.15 \[ -\frac{\, _2F_1\left (-\frac{5}{6},\frac{3}{2};\frac{1}{6};-\frac{x^6}{2}\right )}{10 \sqrt{2} x^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 38, normalized size = 0.2 \begin{align*} -{\frac{4\,{x}^{6}+3}{30\,{x}^{5}}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{2\,x\sqrt{2}}{15}{\mbox{$_2$F$_1$}({\frac{1}{6}},{\frac{1}{2}};\,{\frac{7}{6}};\,-{\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^{6}}\,{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^{18} + 4 \, x^{12} + 4 \, x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.05817, size = 39, normalized size = 0.2 \begin{align*} \frac{\sqrt{2} \Gamma \left (- \frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{6}, \frac{3}{2} \\ \frac{1}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 x^{5} \Gamma \left (\frac{1}{6}\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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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