3.1404 \(\int \frac{x^4}{\sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=376 \[ \frac{\left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{2 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}-\frac{\left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}-\frac{\sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

[Out]

((1 + Sqrt[3])*x*Sqrt[2 + x^6])/(2*(2^(1/3) + (1 + Sqrt[3])*x^2)) - (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])/(2^(2/3)*Sqrt[(x^2*(2^(1/3) + x^2))/(2^(1/3) + (1 + Sqrt[3])
*x^2)^2]*Sqrt[2 + x^6]) - ((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]*EllipticF[ArcCos[(2^(1/3) + (1 -
Sqrt[3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(2*2^(2/3)*3^(1/
4)*Sqrt[(x^2*(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 = 0.201722, antiderivative size = 376, 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 \[ \frac{\left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{2 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}-\frac{\left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}-\frac{\sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[2 + x^6],x]

[Out]

((1 + Sqrt[3])*x*Sqrt[2 + x^6])/(2*(2^(1/3) + (1 + Sqrt[3])*x^2)) - (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])/(2^(2/3)*Sqrt[(x^2*(2^(1/3) + x^2))/(2^(1/3) + (1 + Sqrt[3])
*x^2)^2]*Sqrt[2 + x^6]) - ((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]*EllipticF[ArcCos[(2^(1/3) + (1 -
Sqrt[3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(2*2^(2/3)*3^(1/
4)*Sqrt[(x^2*(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 = 12.5239, size = 338, normalized size = 0.9 \[ - \frac{2^{\frac{2}{3}} \sqrt [4]{3} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (x^{2} + \sqrt [3]{2}\right ) E\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{4 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{3}{4}} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (- 4 \sqrt{3} + 4\right ) \left (x^{2} + \sqrt [3]{2}\right ) F\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{96 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} + \frac{x \left (\frac{1}{2} + \frac{\sqrt{3}}{2}\right ) \sqrt{x^{6} + 2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(x**6+2)**(1/2),x)

[Out]

-2**(2/3)*3**(1/4)*x*sqrt((2*2**(1/3)*x**4 - 2*2**(2/3)*x**2 + 4)/(x**2*(1 + sqr
t(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)/(4*sqrt(x**2*(x**2 +
2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*sqrt(x**6 + 2)) - 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)/(96*sqrt(x**2*(x**2
+ 2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*sqrt(x**6 + 2)) + x*(1/2 + sqrt
(3)/2)*sqrt(x**6 + 2)/(x**2*(1 + sqrt(3)) + 2**(1/3))

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Mathematica [A]  time = 0.654361, size = 276, normalized size = 0.73 \[ \frac{6 \left (1+\sqrt{3}\right ) x^2 \left (x^6+2\right )+\sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^3 \left (-\left (\sqrt{3}-3\right ) F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )-6 E\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )\right )}{12 x \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right ) \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[2 + x^6],x]

[Out]

(6*(1 + Sqrt[3])*x^2*(2 + x^6) + 2^(1/3)*3^(1/4)*Sqrt[(x^2*(2^(1/3) + x^2))/(2^(
1/3) + (1 + Sqrt[3])*x^2)^2]*(2^(1/3) + (1 + Sqrt[3])*x^2)^3*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 + Sq
rt[3])*EllipticF[ArcCos[(2^(1/3) - (-1 + Sqrt[3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*
x^2)], (2 + Sqrt[3])/4]))/(12*x*(2^(1/3) + (1 + Sqrt[3])*x^2)*Sqrt[2 + x^6])

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Maple [C]  time = 0.023, size = 20, normalized size = 0.1 \[{\frac{\sqrt{2}{x}^{5}}{10}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{5}{6}};\,{\frac{11}{6}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^6+2)^(1/2),x)

[Out]

1/10*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. \[ \int \frac{x^{4}}{\sqrt{x^{6} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(x^6 + 2),x, algorithm="maxima")

[Out]

integrate(x^4/sqrt(x^6 + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{\sqrt{x^{6} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(x^6 + 2),x, algorithm="fricas")

[Out]

integral(x^4/sqrt(x^6 + 2), x)

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Sympy [A]  time = 2.00661, size = 36, normalized size = 0.1 \[ \frac{\sqrt{2} x^{5} \Gamma \left (\frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{12 \Gamma \left (\frac{11}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(x**6+2)**(1/2),x)

[Out]

sqrt(2)*x**5*gamma(5/6)*hyper((1/2, 5/6), (11/6,), x**6*exp_polar(I*pi)/2)/(12*g
amma(11/6))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{x^{6} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(x^6 + 2),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(x^6 + 2), x)