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

Optimal. Leaf size=151 \[ \frac{\sqrt{-x^3-1}}{5 x^5}-\frac{7 \sqrt{-x^3-1}}{20 x^2}+\frac{7 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{20 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[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 = 0.085168, antiderivative size = 151, 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 \[ \frac{\sqrt{-x^3-1}}{5 x^5}-\frac{7 \sqrt{-x^3-1}}{20 x^2}+\frac{7 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{20 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

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 = 6.47314, size = 129, normalized size = 0.85 \[ \frac{7 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{60 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} - \frac{7 \sqrt{- x^{3} - 1}}{20 x^{2}} + \frac{\sqrt{- x^{3} - 1}}{5 x^{5}} \]

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

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Mathematica [C]  time = 0.132432, size = 117, normalized size = 0.77 \[ \frac{21 x^6+9 x^3+7 (-1)^{5/6} 3^{3/4} \sqrt{-(-1)^{5/6}+i x} \sqrt{-\sqrt [3]{-1} x^2-(-1)^{2/3} x+1} x^5 F\left (\sin ^{-1}\left (\frac{\sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-12}{60 x^5 \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.034, 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 = 2.77952, size = 39, normalized size = 0.26 \[ - \frac{i \Gamma \left (- \frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{3}, \frac{1}{2} \\ - \frac{2}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 x^{5} \Gamma \left (- \frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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