3.1422 \(\int \frac{x^{12}}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=195 \[ \frac{7}{12} \sqrt{x^6+2} x-\frac{x^7}{3 \sqrt{x^6+2}}-\frac{7 \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

[Out]

-x^7/(3*Sqrt[2 + x^6]) + (7*x*Sqrt[2 + x^6])/12 - (7*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])/(12*
2^(1/3)*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 = 0.128704, antiderivative size = 195, 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 \[ \frac{7}{12} \sqrt{x^6+2} x-\frac{x^7}{3 \sqrt{x^6+2}}-\frac{7 \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^12/(2 + x^6)^(3/2),x]

[Out]

-x^7/(3*Sqrt[2 + x^6]) + (7*x*Sqrt[2 + x^6])/12 - (7*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])/(12*
2^(1/3)*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 = 6.47434, size = 170, normalized size = 0.87 \[ - \frac{x^{7}}{3 \sqrt{x^{6} + 2}} - \frac{7 \cdot 3^{\frac{3}{4}} x \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \left (x^{2} + \sqrt [3]{2}\right ) F\left (\operatorname{acos}{\left (\frac{x^{2} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{2}}{x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{72 \sqrt{\frac{x^{2} \left (x^{2} + \sqrt [3]{2}\right )}{\left (x^{2} \left (1 + \sqrt{3}\right ) + \sqrt [3]{2}\right )^{2}}} \sqrt{x^{6} + 2}} + \frac{7 x \sqrt{x^{6} + 2}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12/(x**6+2)**(3/2),x)

[Out]

-x**7/(3*sqrt(x**6 + 2)) - 7*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)*(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)/(
72*sqrt(x**2*(x**2 + 2**(1/3))/(x**2*(1 + sqrt(3)) + 2**(1/3))**2)*sqrt(x**6 + 2
)) + 7*x*sqrt(x**6 + 2)/12

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Mathematica [A]  time = 0.969947, size = 183, normalized size = 0.94 \[ \frac{6 x^2 \left (3 x^6+14\right )-\frac{7\ 2^{2/3} 3^{3/4} x^2 \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (-1+\sqrt{3}\right ) x^2}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}}}}{72 x \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12/(2 + x^6)^(3/2),x]

[Out]

(6*x^2*(14 + 3*x^6) - (7*2^(2/3)*3^(3/4)*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]*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])/(72*x*Sqrt[2 + x^6])

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Maple [C]  time = 0.035, size = 36, normalized size = 0.2 \[{\frac{x \left ( 3\,{x}^{6}+14 \right ) }{12}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{7\,x\sqrt{2}}{12}{\mbox{$_2$F$_1$}({\frac{1}{6}},{\frac{1}{2}};\,{\frac{7}{6}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12/(x^6+2)^(3/2),x)

[Out]

1/12*x*(3*x^6+14)/(x^6+2)^(1/2)-7/12*2^(1/2)*x*hypergeom([1/6,1/2],[7/6],-1/2*x^
6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{12}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(x^6 + 2)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^12/(x^6 + 2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{12}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(x^6 + 2)^(3/2),x, algorithm="fricas")

[Out]

integral(x^12/(x^6 + 2)^(3/2), x)

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Sympy [A]  time = 6.08071, size = 36, normalized size = 0.18 \[ \frac{\sqrt{2} x^{13} \Gamma \left (\frac{13}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{13}{6} \\ \frac{19}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{19}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*x**13*gamma(13/6)*hyper((3/2, 13/6), (19/6,), x**6*exp_polar(I*pi)/2)/(2
4*gamma(19/6))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{12}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(x^6 + 2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^12/(x^6 + 2)^(3/2), x)