3.1431 \(\int \frac{x^4}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=391 \[ \frac{x^5}{6 \sqrt{x^6+2}}-\frac{\left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{6 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}+\frac{\left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{6\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}+\frac{\left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} 3^{3/4} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

[Out]

x^5/(6*Sqrt[2 + x^6]) - ((1 + Sqrt[3])*x*Sqrt[2 + x^6])/(6*(2^(1/3) + (1 + Sqrt[
3])*x^2)) + (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)*3^(3/4)*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])/(6*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 = 0.257083, antiderivative size = 391, 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 \[ \frac{x^5}{6 \sqrt{x^6+2}}-\frac{\left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{6 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}+\frac{\left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{6\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}+\frac{\left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} 3^{3/4} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(2 + x^6)^(3/2),x]

[Out]

x^5/(6*Sqrt[2 + x^6]) - ((1 + Sqrt[3])*x*Sqrt[2 + x^6])/(6*(2^(1/3) + (1 + Sqrt[
3])*x^2)) + (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)*3^(3/4)*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])/(6*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 = 15.3374, size = 352, normalized size = 0.9 \[ \frac{x^{5}}{6 \sqrt{x^{6} + 2}} + \frac{2^{\frac{2}{3}} \sqrt [4]{3} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (x^{2} + \sqrt [3]{2}\right ) E\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{12 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{3}{4}} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (- 4 \sqrt{3} + 4\right ) \left (x^{2} + \sqrt [3]{2}\right ) F\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{288 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} - \frac{x \left (\frac{1}{6} + \frac{\sqrt{3}}{6}\right ) \sqrt{x^{6} + 2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(x**6+2)**(3/2),x)

[Out]

x**5/(6*sqrt(x**6 + 2)) + 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)/(12*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(ac
os((x**2*(-sqrt(3) + 1) + 2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))), sqrt(3)/4
+ 1/2)/(288*sqrt(x**2*(x**2 + 2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*sqrt
(x**6 + 2)) - x*(1/6 + sqrt(3)/6)*sqrt(x**6 + 2)/(x**2*(1 + sqrt(3)) + 2**(1/3))

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Mathematica [A]  time = 0.734228, size = 273, normalized size = 0.7 \[ \frac{6 x^6-\frac{6 \left (1+\sqrt{3}\right ) \left (x^6+2\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}+\frac{\sqrt [3]{2} \sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x^2 \left (\left (\sqrt{3}-3\right ) F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )+6 E\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )\right )}{\sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}}}}{36 x \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(2 + x^6)^(3/2),x]

[Out]

(6*x^6 - (6*(1 + Sqrt[3])*x^2*(2 + x^6))/(2^(1/3) + (1 + Sqrt[3])*x^2) + (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[ArcCos[(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])/(36*x*Sqrt[2 + x^6])

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Maple [C]  time = 0.035, size = 33, normalized size = 0.1 \[{\frac{{x}^{5}}{6}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{\sqrt{2}{x}^{5}}{30}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{5}{6}};\,{\frac{11}{6}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^6+2)^(3/2),x)

[Out]

1/6*x^5/(x^6+2)^(1/2)-1/30*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. \[ \int \frac{x^{4}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^6 + 2)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^4/(x^6 + 2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^6 + 2)^(3/2),x, algorithm="fricas")

[Out]

integral(x^4/(x^6 + 2)^(3/2), x)

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Sympy [A]  time = 2.05952, size = 36, normalized size = 0.09 \[ \frac{\sqrt{2} x^{5} \Gamma \left (\frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{6}, \frac{3}{2} \\ \frac{11}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{11}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*x**5*gamma(5/6)*hyper((5/6, 3/2), (11/6,), x**6*exp_polar(I*pi)/2)/(24*g
amma(11/6))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^6 + 2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^4/(x^6 + 2)^(3/2), x)