3.1418 \(\int \frac{x^{13}}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=202 \[ -\frac{x^8}{3 \sqrt{x^6+2}}+\frac{8}{15} \sqrt{x^6+2} x^2-\frac{16\ 2^{5/6} \sqrt{2+\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{15 \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

[Out]

-x^8/(3*Sqrt[2 + x^6]) + (8*x^2*Sqrt[2 + x^6])/15 - (16*2^(5/6)*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 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^
2)], -7 - 4*Sqrt[3]])/(15*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 = 0.232819, antiderivative size = 202, 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 \[ -\frac{x^8}{3 \sqrt{x^6+2}}+\frac{8}{15} \sqrt{x^6+2} x^2-\frac{16\ 2^{5/6} \sqrt{2+\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{15 \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^13/(2 + x^6)^(3/2),x]

[Out]

-x^8/(3*Sqrt[2 + x^6]) + (8*x^2*Sqrt[2 + x^6])/15 - (16*2^(5/6)*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 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^
2)], -7 - 4*Sqrt[3]])/(15*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 = 9.49002, size = 190, normalized size = 0.94 \[ - \frac{x^{8}}{3 \sqrt{x^{6} + 2}} + \frac{8 x^{2} \sqrt{x^{6} + 2}}{15} - \frac{16 \cdot 3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (2 x^{2} + 2 \sqrt [3]{2}\right ) F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} x^{2} - 2 \sqrt{3} + 2}{2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{45 \sqrt{\frac{2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13/(x**6+2)**(3/2),x)

[Out]

-x**8/(3*sqrt(x**6 + 2)) + 8*x**2*sqrt(x**6 + 2)/15 - 16*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))*elliptic_f(asin((2**(2/3)*x**2 - 2*sqrt(3) + 2)/(2**(
2/3)*x**2 + 2 + 2*sqrt(3))), -7 - 4*sqrt(3))/(45*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 [C]  time = 0.231441, size = 144, normalized size = 0.71 \[ \frac{9 x^8+48 x^2-16 \sqrt [6]{-1} \sqrt [3]{2} 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (2^{2/3} x^2+2 (-1)^{2/3}\right )} \sqrt{(-1)^{2/3} \sqrt [3]{2} x^4+\sqrt [3]{-1} 2^{2/3} x^2+2} F\left (\sin ^{-1}\left (\frac{\sqrt{\left (-i+\sqrt{3}\right ) \left (2^{2/3} x^2+2\right )}}{2 \sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{45 \sqrt{x^6+2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^13/(2 + x^6)^(3/2),x]

[Out]

(48*x^2 + 9*x^8 - 16*(-1)^(1/6)*2^(1/3)*3^(3/4)*Sqrt[-((-1)^(1/6)*(2*(-1)^(2/3)
+ 2^(2/3)*x^2))]*Sqrt[2 + (-1)^(1/3)*2^(2/3)*x^2 + (-1)^(2/3)*2^(1/3)*x^4]*Ellip
ticF[ArcSin[Sqrt[(-I + Sqrt[3])*(2 + 2^(2/3)*x^2)]/(2*3^(1/4))], (-1)^(1/3)])/(4
5*Sqrt[2 + x^6])

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Maple [C]  time = 0.036, size = 40, normalized size = 0.2 \[{\frac{{x}^{2} \left ( 3\,{x}^{6}+16 \right ) }{15}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{8\,{x}^{2}\sqrt{2}}{15}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{4}{3}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13/(x^6+2)^(3/2),x)

[Out]

1/15*x^2*(3*x^6+16)/(x^6+2)^(1/2)-8/15*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. \[ \int \frac{x^{13}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(x^6 + 2)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^13/(x^6 + 2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{13}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(x^6 + 2)^(3/2),x, algorithm="fricas")

[Out]

integral(x^13/(x^6 + 2)^(3/2), x)

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Sympy [A]  time = 7.36117, size = 36, normalized size = 0.18 \[ \frac{\sqrt{2} x^{14} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**13/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*x**14*gamma(7/3)*hyper((3/2, 7/3), (10/3,), x**6*exp_polar(I*pi)/2)/(24*
gamma(10/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(x^6 + 2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^13/(x^6 + 2)^(3/2), x)