3.468 \(\int \frac{1}{x^6 \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=154 \[ -\frac{\sqrt{1-x^3}}{5 x^5}-\frac{7 \sqrt{1-x^3}}{20 x^2}-\frac{7 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{20 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

[Out]

-Sqrt[1 - x^3]/(5*x^5) - (7*Sqrt[1 - x^3])/(20*x^2) - (7*Sqrt[2 + Sqrt[3]]*(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]])/(20*3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]
*Sqrt[1 - x^3])

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Rubi [A]  time = 0.096367, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{1-x^3}}{5 x^5}-\frac{7 \sqrt{1-x^3}}{20 x^2}-\frac{7 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{20 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*Sqrt[1 - x^3]),x]

[Out]

-Sqrt[1 - x^3]/(5*x^5) - (7*Sqrt[1 - x^3])/(20*x^2) - (7*Sqrt[2 + Sqrt[3]]*(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]])/(20*3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]
*Sqrt[1 - x^3])

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Rubi in Sympy [A]  time = 6.53093, size = 126, normalized size = 0.82 \[ - \frac{7 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{60 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} - \frac{7 \sqrt{- x^{3} + 1}}{20 x^{2}} - \frac{\sqrt{- x^{3} + 1}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*3**(3/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)*(-x + 1
)*elliptic_f(asin((-x - sqrt(3) + 1)/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(60*sq
rt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 + 1)) - 7*sqrt(-x**3 + 1)/(20*x**2
) - sqrt(-x**3 + 1)/(5*x**5)

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Mathematica [C]  time = 0.0726323, size = 95, normalized size = 0.62 \[ \frac{21 x^6-9 x^3+7 i 3^{3/4} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} x^5 F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-12}{60 x^5 \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^6*Sqrt[1 - x^3]),x]

[Out]

(-12 - 9*x^3 + 21*x^6 + (7*I)*3^(3/4)*Sqrt[(-1)^(5/6)*(-1 + x)]*x^5*Sqrt[1 + x +
x^2]*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - I*x]/3^(1/4)], (-1)^(1/3)])/(60*x^5*Sq
rt[1 - x^3])

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Maple [A]  time = 0.035, size = 136, normalized size = 0.9 \[ -{\frac{1}{5\,{x}^{5}}\sqrt{-{x}^{3}+1}}-{\frac{7}{20\,{x}^{2}}\sqrt{-{x}^{3}+1}}-{{\frac{7\,i}{60}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(-x^3+1)^(1/2),x)

[Out]

-1/5*(-x^3+1)^(1/2)/x^5-7/20*(-x^3+1)^(1/2)/x^2-7/60*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)*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. \[ \int \frac{1}{\sqrt{-x^{3} + 1} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-x^3 + 1)*x^6), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(-x^3 + 1)*x^6), x)

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Sympy [A]  time = 2.89434, size = 37, normalized size = 0.24 \[ \frac{\Gamma \left (- \frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{3}, \frac{1}{2} \\ - \frac{2}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 x^{5} \Gamma \left (- \frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-5/3)*hyper((-5/3, 1/2), (-2/3,), x**3*exp_polar(2*I*pi))/(3*x**5*gamma(-2
/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{3} + 1} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-x^3 + 1)*x^6), x)