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

Optimal. Leaf size=131 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{2 \sqrt{2-\sqrt{3}} f (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[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 = 0.126055, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{2 \sqrt{2-\sqrt{3}} f (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

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 = 12.6552, size = 116, normalized size = 0.89 \[ \frac{2 e \operatorname{atan}{\left (\sqrt{- x^{3} - 1} \right )}}{3} + \frac{2 \cdot 3^{\frac{3}{4}} f \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} \]

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 = 0.623961, size = 138, normalized size = 1.05 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{-x^3-1}\right )-\frac{2 f \left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \sqrt{-\frac{(-1)^{2/3} \left (x+(-1)^{2/3}\right )}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}} \sqrt{-x^3-1}} \]

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*((-1)^(1/3) - x)*Sqrt[(1 + x)/(1 + (-1)^(1
/3))]*Sqrt[-(((-1)^(2/3)*((-1)^(2/3) + x))/(1 + (-1)^(1/3)))]*EllipticF[ArcSin[S
qrt[(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 = 122, normalized size = 0.9 \[{-{\frac{2\,i}{3}}f\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}+{\frac{2\,e}{3}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)/x/(-x^3-1)^(1/2),x)

[Out]

-2/3*I*f*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))+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 = 5.39559, size = 46, normalized size = 0.35 \[ \frac{2 i e \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} - \frac{i f x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*I*e*asinh(x**(-3/2))/3 - I*f*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), x**3*exp_p
olar(I*pi))/(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)