3.472 \(\int \frac{1}{x^2 \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=270 \[ -\frac{\sqrt{1-x^3}}{-x+\sqrt{3}+1}-\frac{\sqrt{1-x^3}}{x}-\frac{\sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

[Out]

-(Sqrt[1 - x^3]/(1 + Sqrt[3] - x)) - Sqrt[1 - x^3]/x + (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]])/(2*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*S
qrt[1 - x^3]) - (Sqrt[2]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*Ellipti
cF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(
1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])

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Rubi [A]  time = 0.177862, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt{1-x^3}}{-x+\sqrt{3}+1}-\frac{\sqrt{1-x^3}}{x}-\frac{\sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[1 - x^3]),x]

[Out]

-(Sqrt[1 - x^3]/(1 + Sqrt[3] - x)) - Sqrt[1 - x^3]/x + (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]])/(2*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*S
qrt[1 - x^3]) - (Sqrt[2]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*Ellipti
cF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(
1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])

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Rubi in Sympy [A]  time = 13.8493, size = 211, normalized size = 0.78 \[ - \frac{\sqrt{- x^{3} + 1}}{- x + 1 + \sqrt{3}} + \frac{\sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) E\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{2 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} - \frac{\sqrt{- x^{3} + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**3 + 1)/(-x + 1 + sqrt(3)) + 3**(1/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqr
t(3))**2)*sqrt(-sqrt(3) + 2)*(-x + 1)*elliptic_e(asin((-x - sqrt(3) + 1)/(-x + 1
+ sqrt(3))), -7 - 4*sqrt(3))/(2*sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3
+ 1)) - sqrt(2)*3**(3/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*(-x + 1)*el
liptic_f(asin((-x - sqrt(3) + 1)/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(3*sqrt((-
x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 + 1)) - sqrt(-x**3 + 1)/x

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Mathematica [C]  time = 0.244517, size = 133, normalized size = 0.49 \[ \frac{\frac{3 \left (x^3-1\right )}{x}+(-1)^{2/3} 3^{3/4} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} \left ((-1)^{5/6} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3 \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^2*Sqrt[1 - x^3]),x]

[Out]

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

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Maple [A]  time = 0.033, size = 173, normalized size = 0.6 \[ -{\frac{1}{x}\sqrt{-{x}^{3}+1}}+{{\frac{i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}} \left ( \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) +{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-x^3+1)^(1/2),x)

[Out]

-(-x^3+1)^(1/2)/x+1/3*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))+EllipticF(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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{3} + 1} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-x^3 + 1)*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(-x^3 + 1)*x^2), x)

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Sympy [A]  time = 2.14919, size = 32, normalized size = 0.12 \[ \frac{\Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{2} \\ \frac{2}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 x \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{3} + 1} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-x^3 + 1)*x^2), x)