3.1426 \(\int \frac{x^{15}}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=394 \[ -\frac{x^{10}}{3 \sqrt{x^6+2}}+\frac{10}{21} \sqrt{x^6+2} x^4-\frac{80 \sqrt{x^6+2}}{21 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{80\ 2^{2/3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{21 \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}+\frac{40 \sqrt [6]{2} \sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{7\ 3^{3/4} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

[Out]

-x^10/(3*Sqrt[2 + x^6]) + (10*x^4*Sqrt[2 + x^6])/21 - (80*Sqrt[2 + x^6])/(21*(2^
(1/3)*(1 + Sqrt[3]) + x^2)) + (40*2^(1/6)*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]*EllipticE[ArcSin
[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/
(7*3^(3/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6])
- (80*2^(2/3)*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2 + x^4)/(2^(1/3)*(1 + S
qrt[3]) + x^2)^2]*EllipticF[ArcSin[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + S
qrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(21*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 = 0.493421, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{x^{10}}{3 \sqrt{x^6+2}}+\frac{10}{21} \sqrt{x^6+2} x^4-\frac{80 \sqrt{x^6+2}}{21 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{80\ 2^{2/3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{21 \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}+\frac{40 \sqrt [6]{2} \sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{7\ 3^{3/4} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^15/(2 + x^6)^(3/2),x]

[Out]

-x^10/(3*Sqrt[2 + x^6]) + (10*x^4*Sqrt[2 + x^6])/21 - (80*Sqrt[2 + x^6])/(21*(2^
(1/3)*(1 + Sqrt[3]) + x^2)) + (40*2^(1/6)*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]*EllipticE[ArcSin
[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/
(7*3^(3/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6])
- (80*2^(2/3)*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2 + x^4)/(2^(1/3)*(1 + S
qrt[3]) + x^2)^2]*EllipticF[ArcSin[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + S
qrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(21*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 [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**15/(x**6+2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [C]  time = 0.540099, size = 195, normalized size = 0.49 \[ \frac{3 \left (3 x^6+20\right ) x^4+40\ 2^{2/3} 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (2^{2/3} x^2+2 (-1)^{2/3}\right )} \sqrt{(-1)^{2/3} \sqrt [3]{2} x^4+\sqrt [3]{-1} 2^{2/3} x^2+2} \left ((-1)^{5/6} F\left (\sin ^{-1}\left (\frac{\sqrt{\left (-i+\sqrt{3}\right ) \left (2^{2/3} x^2+2\right )}}{2 \sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{\left (-i+\sqrt{3}\right ) \left (2^{2/3} x^2+2\right )}}{2 \sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{63 \sqrt{x^6+2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^15/(2 + x^6)^(3/2),x]

[Out]

(3*x^4*(20 + 3*x^6) + 40*2^(2/3)*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]*(Sqrt[3]*Ell
ipticE[ArcSin[Sqrt[(-I + Sqrt[3])*(2 + 2^(2/3)*x^2)]/(2*3^(1/4))], (-1)^(1/3)] +
(-1)^(5/6)*EllipticF[ArcSin[Sqrt[(-I + Sqrt[3])*(2 + 2^(2/3)*x^2)]/(2*3^(1/4))]
, (-1)^(1/3)]))/(63*Sqrt[2 + x^6])

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Maple [C]  time = 0.035, size = 40, normalized size = 0.1 \[{\frac{{x}^{4} \left ( 3\,{x}^{6}+20 \right ) }{21}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{10\,{x}^{4}\sqrt{2}}{21}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{2}{3}};\,{\frac{5}{3}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^15/(x^6+2)^(3/2),x)

[Out]

1/21*x^4*(3*x^6+20)/(x^6+2)^(1/2)-10/21*2^(1/2)*x^4*hypergeom([1/2,2/3],[5/3],-1
/2*x^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{15}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/(x^6 + 2)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^15/(x^6 + 2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{15}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/(x^6 + 2)^(3/2),x, algorithm="fricas")

[Out]

integral(x^15/(x^6 + 2)^(3/2), x)

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Sympy [A]  time = 10.8667, size = 36, normalized size = 0.09 \[ \frac{\sqrt{2} x^{16} \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{11}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**15/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*x**16*gamma(8/3)*hyper((3/2, 8/3), (11/3,), x**6*exp_polar(I*pi)/2)/(24*
gamma(11/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{15}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/(x^6 + 2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^15/(x^6 + 2)^(3/2), x)