3.473 \(\int \frac{1}{x^5 \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=294 \[ -\frac{5 \sqrt{1-x^3}}{8 \left (-x+\sqrt{3}+1\right )}-\frac{5 \sqrt{1-x^3}}{8 x}-\frac{\sqrt{1-x^3}}{4 x^4}-\frac{5 (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}+\frac{5 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{16 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

[Out]

(-5*Sqrt[1 - x^3])/(8*(1 + Sqrt[3] - x)) - Sqrt[1 - x^3]/(4*x^4) - (5*Sqrt[1 - x
^3])/(8*x) + (5*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3
] - x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]]
)/(16*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3]) - (5*(1 - x)*Sqrt[(1 + x
+ x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)
], -7 - 4*Sqrt[3]])/(4*Sqrt[2]*3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1
- x^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.217124, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{5 \sqrt{1-x^3}}{8 \left (-x+\sqrt{3}+1\right )}-\frac{5 \sqrt{1-x^3}}{8 x}-\frac{\sqrt{1-x^3}}{4 x^4}-\frac{5 (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}+\frac{5 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{16 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[1 - x^3]),x]

[Out]

(-5*Sqrt[1 - x^3])/(8*(1 + Sqrt[3] - x)) - Sqrt[1 - x^3]/(4*x^4) - (5*Sqrt[1 - x
^3])/(8*x) + (5*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3
] - x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]]
)/(16*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3]) - (5*(1 - x)*Sqrt[(1 + x
+ x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)
], -7 - 4*Sqrt[3]])/(4*Sqrt[2]*3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1
- x^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.4198, size = 236, normalized size = 0.8 \[ - \frac{5 \sqrt{- x^{3} + 1}}{8 \left (- x + 1 + \sqrt{3}\right )} + \frac{5 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) E\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{16 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} - \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{24 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} - \frac{5 \sqrt{- x^{3} + 1}}{8 x} - \frac{\sqrt{- x^{3} + 1}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(-x**3+1)**(1/2),x)

[Out]

-5*sqrt(-x**3 + 1)/(8*(-x + 1 + sqrt(3))) + 5*3**(1/4)*sqrt((x**2 + x + 1)/(-x +
1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(-x + 1)*elliptic_e(asin((-x - sqrt(3) + 1)
/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(16*sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*s
qrt(-x**3 + 1)) - 5*sqrt(2)*3**(3/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*
(-x + 1)*elliptic_f(asin((-x - sqrt(3) + 1)/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))
/(24*sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 + 1)) - 5*sqrt(-x**3 + 1)/(
8*x) - sqrt(-x**3 + 1)/(4*x**4)

_______________________________________________________________________________________

Mathematica [C]  time = 0.257495, size = 145, normalized size = 0.49 \[ \frac{3 \left (x^3-1\right ) \left (5 x^3+2\right )+\frac{5\ 3^{3/4} (x-1) \sqrt{x^2+x+1} x^4 \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{\sqrt{(-1)^{5/6} (x-1)}}}{24 x^4 \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^5*Sqrt[1 - x^3]),x]

[Out]

(3*(-1 + x^3)*(2 + 5*x^3) + (5*3^(3/4)*(-1 + x)*x^4*Sqrt[1 + x + x^2]*((-I)*Sqrt
[3]*EllipticE[ArcSin[Sqrt[-(-1)^(5/6) - I*x]/3^(1/4)], (-1)^(1/3)] + (-1)^(1/3)*
EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - I*x]/3^(1/4)], (-1)^(1/3)]))/Sqrt[(-1)^(5/6)
*(-1 + x)])/(24*x^4*Sqrt[1 - x^3])

_______________________________________________________________________________________

Maple [A]  time = 0.035, size = 187, normalized size = 0.6 \[ -{\frac{1}{4\,{x}^{4}}\sqrt{-{x}^{3}+1}}-{\frac{5}{8\,x}\sqrt{-{x}^{3}+1}}+{{\frac{5\,i}{24}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}} \left ( \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) +{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(-x^3+1)^(1/2),x)

[Out]

-1/4*(-x^3+1)^(1/2)/x^4-5/8*(-x^3+1)^(1/2)/x+5/24*I*3^(1/2)*(I*(x+1/2-1/2*I*3^(1
/2))*3^(1/2))^(1/2)*((-1+x)/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1/2)
)*3^(1/2))^(1/2)/(-x^3+1)^(1/2)*((-3/2+1/2*I*3^(1/2))*EllipticE(1/3*3^(1/2)*(I*(
x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))+Elli
pticF(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(-3/2+1/2*I
*3^(1/2)))^(1/2)))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{3} + 1} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^3 + 1)*x^5),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^3 + 1)*x^5), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{3} + 1} x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^3 + 1)*x^5),x, algorithm="fricas")

[Out]

integral(1/(sqrt(-x^3 + 1)*x^5), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.6333, size = 37, normalized size = 0.13 \[ \frac{\Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{2} \\ - \frac{1}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 x^{4} \Gamma \left (- \frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(-x**3+1)**(1/2),x)

[Out]

gamma(-4/3)*hyper((-4/3, 1/2), (-1/3,), x**3*exp_polar(2*I*pi))/(3*x**4*gamma(-1
/3))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{3} + 1} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^3 + 1)*x^5),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^3 + 1)*x^5), x)