Optimal. Leaf size=166 \[ \frac{\sqrt{2+\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{\sqrt [6]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]
[Out]
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Rubi [A] time = 0.143609, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\sqrt{2+\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{\sqrt [6]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
[In] Int[x/Sqrt[2 + x^6],x]
[Out]
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Rubi in Sympy [A] time = 4.36972, size = 160, normalized size = 0.96 \[ \frac{3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (2 x^{2} + 2 \sqrt [3]{2}\right ) F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} x^{2} - 2 \sqrt{3} + 2}{2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{6 \sqrt{\frac{2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{x^{6} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(x**6+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.156012, size = 116, normalized size = 0.7 \[ \frac{\sqrt [6]{-1} \sqrt [3]{2} \sqrt{(-1)^{5/6} \left (\sqrt [3]{-\frac{1}{2}} x^2-1\right )} \sqrt{\left (-\frac{1}{2}\right )^{2/3} x^4+\sqrt [3]{-\frac{1}{2}} x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{(-1)^{5/6} x^2}{\sqrt [3]{2}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt{x^6+2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x/Sqrt[2 + x^6],x]
[Out]
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Maple [C] time = 0.023, size = 20, normalized size = 0.1 \[{\frac{{x}^{2}\sqrt{2}}{4}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{4}{3}};\,-{\frac{{x}^{6}}{2}})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(x^6+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{6} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(x^6 + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{x^{6} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(x^6 + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.77284, size = 36, normalized size = 0.22 \[ \frac{\sqrt{2} x^{2} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{12 \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x**6+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{6} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(x^6 + 2),x, algorithm="giac")
[Out]