3.1429 \(\int \frac{1}{x^3 \left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=394 \[ \frac{5 \sqrt{x^6+2}}{24 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{5 \sqrt{x^6+2}}{24 x^2}+\frac{1}{6 x^2 \sqrt{x^6+2}}+\frac{5 \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}-\frac{5 \sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{8\ 2^{5/6} 3^{3/4} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

[Out]

1/(6*x^2*Sqrt[2 + x^6]) - (5*Sqrt[2 + x^6])/(24*x^2) + (5*Sqrt[2 + x^6])/(24*(2^
(1/3)*(1 + Sqrt[3]) + x^2)) - (5*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]])/(8*2^(5/6
)*3^(3/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6]) +
(5*(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]])/(12*2^(1/3)*3^(1/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + S
qrt[3]) + x^2)^2]*Sqrt[2 + x^6])

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Rubi [A]  time = 0.491701, 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 \[ \frac{5 \sqrt{x^6+2}}{24 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{5 \sqrt{x^6+2}}{24 x^2}+\frac{1}{6 x^2 \sqrt{x^6+2}}+\frac{5 \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}-\frac{5 \sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{8\ 2^{5/6} 3^{3/4} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(2 + x^6)^(3/2)),x]

[Out]

1/(6*x^2*Sqrt[2 + x^6]) - (5*Sqrt[2 + x^6])/(24*x^2) + (5*Sqrt[2 + x^6])/(24*(2^
(1/3)*(1 + Sqrt[3]) + x^2)) - (5*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]])/(8*2^(5/6
)*3^(3/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6]) +
(5*(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]])/(12*2^(1/3)*3^(1/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + S
qrt[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(1/x**3/(x**6+2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [C]  time = 0.866088, size = 198, normalized size = 0.5 \[ \frac{i \left (6 i x^6+9 i \left (x^6+2\right )+5 i 2^{2/3} 3^{3/4} \sqrt{(-1)^{5/6} \left (\sqrt [3]{-\frac{1}{2}} x^2-1\right )} \sqrt{\left (-\frac{1}{2}\right )^{2/3} x^4+\sqrt [3]{-\frac{1}{2}} x^2+1} x^2 \left ((-1)^{5/6} F\left (\sin ^{-1}\left (\frac{\sqrt{\left (-i+\sqrt{3}\right ) \left (2^{2/3} x^2+2\right )}}{2 \sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{\left (-i+\sqrt{3}\right ) \left (2^{2/3} x^2+2\right )}}{2 \sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )}{72 x^2 \sqrt{x^6+2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^3*(2 + x^6)^(3/2)),x]

[Out]

((I/72)*((6*I)*x^6 + (9*I)*(2 + x^6) + (5*I)*2^(2/3)*3^(3/4)*x^2*Sqrt[(-1)^(5/6)
*(-1 + (-1/2)^(1/3)*x^2)]*Sqrt[1 + (-1/2)^(1/3)*x^2 + (-1/2)^(2/3)*x^4]*(Sqrt[3]
*EllipticE[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)])))/(x^2*Sqrt[2 + x^6])

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Maple [C]  time = 0.042, size = 40, normalized size = 0.1 \[ -{\frac{5\,{x}^{6}+6}{24\,{x}^{2}}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{5\,{x}^{4}\sqrt{2}}{192}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{2}{3}};\,{\frac{5}{3}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^6+2)^(3/2),x)

[Out]

-1/24*(5*x^6+6)/x^2/(x^6+2)^(1/2)+5/192*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. \[ \int \frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^3),x, algorithm="maxima")

[Out]

integrate(1/((x^6 + 2)^(3/2)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{9} + 2 \, x^{3}\right )} \sqrt{x^{6} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^3),x, algorithm="fricas")

[Out]

integral(1/((x^9 + 2*x^3)*sqrt(x^6 + 2)), x)

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Sympy [A]  time = 2.62887, size = 39, normalized size = 0.1 \[ \frac{\sqrt{2} \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{3}{2} \\ \frac{2}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 x^{2} \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*gamma(-1/3)*hyper((-1/3, 3/2), (2/3,), x**6*exp_polar(I*pi)/2)/(24*x**2*
gamma(2/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^3),x, algorithm="giac")

[Out]

integrate(1/((x^6 + 2)^(3/2)*x^3), x)