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

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

[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])

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Rubi [A]  time = 0.202563, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{5 \sqrt{x^3-1}}{8 \left (-x-\sqrt{3}+1\right )}+\frac{5 \sqrt{x^3-1}}{8 x}+\frac{\sqrt{x^3-1}}{4 x^4}+\frac{5 (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{5 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16 \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

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])

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Rubi in Sympy [A]  time = 14.636, size = 233, normalized size = 0.79 \[ \frac{5 \sqrt{x^{3} - 1}}{8 \left (- x - \sqrt{3} + 1\right )} - \frac{5 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) E\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{16 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} + \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{24 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} + \frac{5 \sqrt{x^{3} - 1}}{8 x} + \frac{\sqrt{x^{3} - 1}}{4 x^{4}} \]

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

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Mathematica [C]  time = 0.261684, size = 140, normalized size = 0.48 \[ \frac{\frac{3 \left (x^3-1\right ) \left (5 x^3+2\right )}{x^4}+\frac{5\ 3^{3/4} (x-1) \sqrt{x^2+x+1} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{\sqrt{(-1)^{5/6} (x-1)}}}{24 \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((3*(-1 + x^3)*(2 + 5*x^3))/x^4 + (5*3^(3/4)*(-1 + x)*Sqrt[1 + x + x^2]*((-I)*Sq
rt[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*Sqrt[-1 + x^3])

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Maple [A]  time = 0.027, size = 198, normalized size = 0.7 \[{\frac{1}{4\,{x}^{4}}\sqrt{{x}^{3}-1}}+{\frac{5}{8\,x}\sqrt{{x}^{3}-1}}-{\frac{-{\frac{15}{2}}-{\frac{5\,i}{2}}\sqrt{3}}{8}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }} \left ( \left ({\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) + \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \]

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-5/8*(-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)*((3/2-1/2*I*3^(1/2))*El
lipticE(((-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))+(-1/2+1/2*I*3^(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. \[ \int \frac{1}{\sqrt{x^{3} - 1} x^{5}}\,{d x} \]

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)

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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)

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Sympy [A]  time = 2.61475, size = 34, normalized size = 0.12 \[ - \frac{i \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{2} \\ - \frac{1}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 x^{4} \Gamma \left (- \frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)