Optimal. Leaf size=394 \[ \frac{5 \left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x \text{EllipticF}\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{8\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}+\frac{1}{8} \sqrt{x^6+2} x^5-\frac{5 \left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{8 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}+\frac{5 \sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4\ 2^{2/3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]
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Rubi [A] time = 0.130567, antiderivative size = 394, 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 = {321, 308, 225, 1881} \[ \frac{1}{8} \sqrt{x^6+2} x^5-\frac{5 \left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{8 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}+\frac{5 \left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{8\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}+\frac{5 \sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4\ 2^{2/3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 308
Rule 225
Rule 1881
Rubi steps
\begin{align*} \int \frac{x^{10}}{\sqrt{2+x^6}} \, dx &=\frac{1}{8} x^5 \sqrt{2+x^6}-\frac{5}{4} \int \frac{x^4}{\sqrt{2+x^6}} \, dx\\ &=\frac{1}{8} x^5 \sqrt{2+x^6}+\frac{5}{8} \int \frac{2^{2/3} \left (-1+\sqrt{3}\right )-2 x^4}{\sqrt{2+x^6}} \, dx+\frac{\left (5 \left (1-\sqrt{3}\right )\right ) \int \frac{1}{\sqrt{2+x^6}} \, dx}{4 \sqrt [3]{2}}\\ &=\frac{1}{8} x^5 \sqrt{2+x^6}-\frac{5 \left (1+\sqrt{3}\right ) x \sqrt{2+x^6}}{8 \left (\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2\right )}+\frac{5 \sqrt [4]{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}} E\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 )}{4\ 2^{2/3} \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}}+\frac{5 \left (1-\sqrt{3}\right ) 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 )}{8\ 2^{2/3} \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 = 0.0067514, size = 41, normalized size = 0.1 \[ \frac{1}{8} x^5 \left (\sqrt{x^6+2}-\sqrt{2} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};-\frac{x^6}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.032, size = 33, normalized size = 0.1 \begin{align*}{\frac{{x}^{5}}{8}\sqrt{{x}^{6}+2}}-{\frac{{x}^{5}\sqrt{2}}{8}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{5}{6}};\,{\frac{11}{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{x^{10}}{\sqrt{x^{6} + 2}}\,{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{x^{10}}{\sqrt{x^{6} + 2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.970959, size = 36, normalized size = 0.09 \begin{align*} \frac{\sqrt{2} x^{11} \Gamma \left (\frac{11}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{6} \\ \frac{17}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{12 \Gamma \left (\frac{17}{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{x^{10}}{\sqrt{x^{6} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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