3.147 \(\int \frac{e+f x}{x \sqrt{-1+x^3}} \, dx\)

Optimal. Leaf size=137 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{x^3-1}\right )-\frac{2 \sqrt{2-\sqrt{3}} f (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

[Out]

(2*e*ArcTan[Sqrt[-1 + x^3]])/3 - (2*Sqrt[2 - Sqrt[3]]*f*(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]])/(3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

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Rubi [A]  time = 0.121515, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{x^3-1}\right )-\frac{2 \sqrt{2-\sqrt{3}} f (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[(e + f*x)/(x*Sqrt[-1 + x^3]),x]

[Out]

(2*e*ArcTan[Sqrt[-1 + x^3]])/3 - (2*Sqrt[2 - Sqrt[3]]*f*(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]])/(3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

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Rubi in Sympy [A]  time = 11.4041, size = 110, normalized size = 0.8 \[ \frac{2 e \operatorname{atan}{\left (\sqrt{x^{3} - 1} \right )}}{3} - \frac{2 \cdot 3^{\frac{3}{4}} f \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*atan(sqrt(x**3 - 1))/3 - 2*3**(3/4)*f*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))/(3*sqrt((x - 1)/(-x - sqrt(3) + 1)**2)*sqrt(x**3 - 1))

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Mathematica [A]  time = 0.490426, size = 136, normalized size = 0.99 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{x^3-1}\right )+\frac{2 f \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \sqrt{x^3-1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*ArcTan[Sqrt[-1 + x^3]])/3 + (2*f*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((-1)^(1/3)
+ x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(
1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(Sqrt[(1 - (-1)^(2/3)*x)/(1 +
(-1)^(1/3))]*Sqrt[-1 + x^3])

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Maple [A]  time = 0.008, size = 129, normalized size = 0.9 \[ 2\,{\frac{f \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }+{\frac{2\,e}{3}\arctan \left ( \sqrt{{x}^{3}-1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)/x/(x^3-1)^(1/2),x)

[Out]

2*f*(-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))+2/3*e*arctan((x^3-1)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)/(sqrt(x^3 - 1)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((f*x + e)/(sqrt(x^3 - 1)*x), x)

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Sympy [A]  time = 5.49916, size = 60, normalized size = 0.44 \[ e \left (\begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{otherwise} \end{cases}\right ) - \frac{i f x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*Piecewise((2*I*acosh(x**(-3/2))/3, Abs(x**(-3)) > 1), (-2*asin(x**(-3/2))/3, T
rue)) - I*f*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), x**3)/(3*gamma(4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)/(sqrt(x^3 - 1)*x), x)