3.1403 \(\int \frac{x^{10}}{\sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=394 \[ \frac{1}{8} \sqrt{x^6+2} x^5-\frac{5 \left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{8 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}+\frac{5 \left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{8\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}+\frac{5 \sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4\ 2^{2/3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

[Out]

(x^5*Sqrt[2 + x^6])/8 - (5*(1 + Sqrt[3])*x*Sqrt[2 + x^6])/(8*(2^(1/3) + (1 + Sqr
t[3])*x^2)) + (5*3^(1/4)*x*(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]*EllipticE[ArcCos[(2^(1/3) + (1 - Sqrt[3])*x^2)/(2
^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(4*2^(2/3)*Sqrt[(x^2*(2^(1/3) +
x^2))/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*Sqrt[2 + x^6]) + (5*(1 - Sqrt[3])*x*(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]*E
llipticF[ArcCos[(2^(1/3) + (1 - Sqrt[3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2
+ Sqrt[3])/4])/(8*2^(2/3)*3^(1/4)*Sqrt[(x^2*(2^(1/3) + x^2))/(2^(1/3) + (1 + Sq
rt[3])*x^2)^2]*Sqrt[2 + x^6])

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Rubi [A]  time = 0.288599, 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 \[ \frac{1}{8} \sqrt{x^6+2} x^5-\frac{5 \left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{8 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}+\frac{5 \left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{8\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}+\frac{5 \sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4\ 2^{2/3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^10/Sqrt[2 + x^6],x]

[Out]

(x^5*Sqrt[2 + x^6])/8 - (5*(1 + Sqrt[3])*x*Sqrt[2 + x^6])/(8*(2^(1/3) + (1 + Sqr
t[3])*x^2)) + (5*3^(1/4)*x*(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]*EllipticE[ArcCos[(2^(1/3) + (1 - Sqrt[3])*x^2)/(2
^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(4*2^(2/3)*Sqrt[(x^2*(2^(1/3) +
x^2))/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*Sqrt[2 + x^6]) + (5*(1 - Sqrt[3])*x*(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]*E
llipticF[ArcCos[(2^(1/3) + (1 - Sqrt[3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2
+ Sqrt[3])/4])/(8*2^(2/3)*3^(1/4)*Sqrt[(x^2*(2^(1/3) + x^2))/(2^(1/3) + (1 + Sq
rt[3])*x^2)^2]*Sqrt[2 + x^6])

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Rubi in Sympy [A]  time = 15.1857, size = 357, normalized size = 0.91 \[ \frac{x^{5} \sqrt{x^{6} + 2}}{8} + \frac{5 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (x^{2} + \sqrt [3]{2}\right ) E\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{16 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} + \frac{5 \cdot 2^{\frac{2}{3}} \cdot 3^{\frac{3}{4}} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (- 4 \sqrt{3} + 4\right ) \left (x^{2} + \sqrt [3]{2}\right ) F\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{384 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} - \frac{x \left (\frac{5}{8} + \frac{5 \sqrt{3}}{8}\right ) \sqrt{x^{6} + 2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(x**6+2)**(1/2),x)

[Out]

x**5*sqrt(x**6 + 2)/8 + 5*2**(2/3)*3**(1/4)*x*sqrt((2*2**(1/3)*x**4 - 2*2**(2/3)
*x**2 + 4)/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*(x**2 + 2**(1/3))*elliptic_e(acos
((x**2*(-sqrt(3) + 1) + 2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))), sqrt(3)/4 +
1/2)/(16*sqrt(x**2*(x**2 + 2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*sqrt(x*
*6 + 2)) + 5*2**(2/3)*3**(3/4)*x*sqrt((2*2**(1/3)*x**4 - 2*2**(2/3)*x**2 + 4)/(x
**2*(1 + sqrt(3)) + 2**(1/3))**2)*(-4*sqrt(3) + 4)*(x**2 + 2**(1/3))*elliptic_f(
acos((x**2*(-sqrt(3) + 1) + 2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))), sqrt(3)/
4 + 1/2)/(384*sqrt(x**2*(x**2 + 2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*sq
rt(x**6 + 2)) - x*(5/8 + 5*sqrt(3)/8)*sqrt(x**6 + 2)/(x**2*(1 + sqrt(3)) + 2**(1
/3))

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Mathematica [A]  time = 0.83854, size = 279, normalized size = 0.71 \[ \frac{6 \left (x^6+2\right ) x^6-\frac{30 \left (1+\sqrt{3}\right ) \left (x^6+2\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}+\frac{5 \sqrt [3]{2} \sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x^2 \left (\left (\sqrt{3}-3\right ) F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )+6 E\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )\right )}{\sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}}}}{48 x \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/Sqrt[2 + x^6],x]

[Out]

(6*x^6*(2 + x^6) - (30*(1 + Sqrt[3])*x^2*(2 + x^6))/(2^(1/3) + (1 + Sqrt[3])*x^2
) + (5*2^(1/3)*3^(1/4)*x^2*(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]*(6*EllipticE[ArcCos[(2^(1/3) - (-1 + Sqrt[3])*x^2
)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4] + (-3 + Sqrt[3])*EllipticF[Ar
cCos[(2^(1/3) - (-1 + Sqrt[3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3]
)/4]))/Sqrt[(x^2*(2^(1/3) + x^2))/(2^(1/3) + (1 + Sqrt[3])*x^2)^2])/(48*x*Sqrt[2
+ x^6])

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Maple [C]  time = 0.049, size = 33, normalized size = 0.1 \[{\frac{{x}^{5}}{8}\sqrt{{x}^{6}+2}}-{\frac{\sqrt{2}{x}^{5}}{8}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{5}{6}};\,{\frac{11}{6}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(x^6+2)^(1/2),x)

[Out]

1/8*x^5*(x^6+2)^(1/2)-1/8*2^(1/2)*x^5*hypergeom([1/2,5/6],[11/6],-1/2*x^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{\sqrt{x^{6} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(x^6 + 2),x, algorithm="maxima")

[Out]

integrate(x^10/sqrt(x^6 + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{10}}{\sqrt{x^{6} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(x^6 + 2),x, algorithm="fricas")

[Out]

integral(x^10/sqrt(x^6 + 2), x)

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Sympy [A]  time = 3.76557, size = 36, normalized size = 0.09 \[ \frac{\sqrt{2} x^{11} \Gamma \left (\frac{11}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{6} \\ \frac{17}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{12 \Gamma \left (\frac{17}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10/(x**6+2)**(1/2),x)

[Out]

sqrt(2)*x**11*gamma(11/6)*hyper((1/2, 11/6), (17/6,), x**6*exp_polar(I*pi)/2)/(1
2*gamma(17/6))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{\sqrt{x^{6} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(x^6 + 2),x, algorithm="giac")

[Out]

integrate(x^10/sqrt(x^6 + 2), x)