3.1420 \(\int \frac{x}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=186 \[ \frac{x^2}{6 \sqrt{x^6+2}}+\frac{\sqrt{2+\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{6 \sqrt [6]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

[Out]

x^2/(6*Sqrt[2 + x^6]) + (Sqrt[2 + Sqrt[3]]*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/
3)*x^2 + x^4)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*EllipticF[ArcSin[(2^(1/3)*(1 - Sq
rt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(6*2^(1/6)*3^(1/4
)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6])

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Rubi [A]  time = 0.183227, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{x^2}{6 \sqrt{x^6+2}}+\frac{\sqrt{2+\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{6 \sqrt [6]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x/(2 + x^6)^(3/2),x]

[Out]

x^2/(6*Sqrt[2 + x^6]) + (Sqrt[2 + Sqrt[3]]*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/
3)*x^2 + x^4)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*EllipticF[ArcSin[(2^(1/3)*(1 - Sq
rt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(6*2^(1/6)*3^(1/4
)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6])

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Rubi in Sympy [A]  time = 5.4895, size = 173, normalized size = 0.93 \[ \frac{x^{2}}{6 \sqrt{x^{6} + 2}} + \frac{3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (2 x^{2} + 2 \sqrt [3]{2}\right ) F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} x^{2} - 2 \sqrt{3} + 2}{2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{36 \sqrt{\frac{2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**6+2)**(3/2),x)

[Out]

x**2/(6*sqrt(x**6 + 2)) + 3**(3/4)*sqrt((2*2**(1/3)*x**4 - 2*2**(2/3)*x**2 + 4)/
(2**(2/3)*x**2 + 2 + 2*sqrt(3))**2)*sqrt(sqrt(3) + 2)*(2*x**2 + 2*2**(1/3))*elli
ptic_f(asin((2**(2/3)*x**2 - 2*sqrt(3) + 2)/(2**(2/3)*x**2 + 2 + 2*sqrt(3))), -7
- 4*sqrt(3))/(36*sqrt((2*2**(2/3)*x**2 + 4)/(2**(2/3)*x**2 + 2 + 2*sqrt(3))**2)
*sqrt(x**6 + 2))

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Mathematica [A]  time = 0.226215, size = 136, normalized size = 0.73 \[ \frac{x^2}{6 \sqrt{x^6+2}}+\frac{\sqrt [6]{-1} \sqrt{(-1)^{5/6} \left (\sqrt [3]{-\frac{1}{2}} x^2-1\right )} \sqrt{\left (-\frac{1}{2}\right )^{2/3} x^4+\sqrt [3]{-\frac{1}{2}} x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{(-1)^{5/6} x^2}{\sqrt [3]{2}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{3\ 2^{2/3} \sqrt [4]{3} \sqrt{x^6+2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/(2 + x^6)^(3/2),x]

[Out]

x^2/(6*Sqrt[2 + x^6]) + ((-1)^(1/6)*Sqrt[(-1)^(5/6)*(-1 + (-1/2)^(1/3)*x^2)]*Sqr
t[1 + (-1/2)^(1/3)*x^2 + (-1/2)^(2/3)*x^4]*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (
(-1)^(5/6)*x^2)/2^(1/3)]/3^(1/4)], (-1)^(1/3)])/(3*2^(2/3)*3^(1/4)*Sqrt[2 + x^6]
)

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Maple [C]  time = 0.034, size = 33, normalized size = 0.2 \[{\frac{{x}^{2}}{6}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{{x}^{2}\sqrt{2}}{24}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{4}{3}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^6+2)^(3/2),x)

[Out]

1/6*x^2/(x^6+2)^(1/2)+1/24*2^(1/2)*x^2*hypergeom([1/3,1/2],[4/3],-1/2*x^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^6 + 2)^(3/2),x, algorithm="maxima")

[Out]

integrate(x/(x^6 + 2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^6 + 2)^(3/2),x, algorithm="fricas")

[Out]

integral(x/(x^6 + 2)^(3/2), x)

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Sympy [A]  time = 2.08163, size = 36, normalized size = 0.19 \[ \frac{\sqrt{2} x^{2} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*x**2*gamma(1/3)*hyper((1/3, 3/2), (4/3,), x**6*exp_polar(I*pi)/2)/(24*ga
mma(4/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^6 + 2)^(3/2),x, algorithm="giac")

[Out]

integrate(x/(x^6 + 2)^(3/2), x)