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

Optimal. Leaf size=155 \[ \frac{\sqrt{x^3-1}}{5 x^5}+\frac{7 \sqrt{x^3-1}}{20 x^2}-\frac{7 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{20 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

[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 = 0.0925096, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{x^3-1}}{5 x^5}+\frac{7 \sqrt{x^3-1}}{20 x^2}-\frac{7 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{20 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

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 = 5.86791, size = 122, normalized size = 0.79 \[ - \frac{7 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{60 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} + \frac{7 \sqrt{x^{3} - 1}}{20 x^{2}} + \frac{\sqrt{x^{3} - 1}}{5 x^{5}} \]

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*s
qrt((x - 1)/(-x - sqrt(3) + 1)**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 = 0.0738946, size = 93, normalized size = 0.6 \[ \frac{21 x^6-9 x^3+7 i 3^{3/4} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} x^5 F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-12}{60 x^5 \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-12 - 9*x^3 + 21*x^6 + (7*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 = 0.028, size = 141, normalized size = 0.9 \[{\frac{1}{5\,{x}^{5}}\sqrt{{x}^{3}-1}}+{\frac{7}{20\,{x}^{2}}\sqrt{{x}^{3}-1}}+{\frac{-{\frac{21}{2}}-{\frac{7\,i}{2}}\sqrt{3}}{20}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \]

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+7/20*(-3/2-1/2*I*3^(1/2))*((-1+x)/(
-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x
+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticF(((-1+x)/(
-3/2-1/2*I*3^(1/2)))^(1/2),((3/2+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 = 2.88296, size = 34, normalized size = 0.22 \[ - \frac{i \Gamma \left (- \frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{3}, \frac{1}{2} \\ - \frac{2}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 x^{5} \Gamma \left (- \frac{2}{3}\right )} \]

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)/(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)