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

Optimal. Leaf size=134 \[ -\frac{2}{3} e \tanh ^{-1}\left (\sqrt{1-x^3}\right )-\frac{2 \sqrt{2+\sqrt{3}} f (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

[Out]

(-2*e*ArcTanh[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 = 0.120678, antiderivative size = 134, 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 \[ -\frac{2}{3} e \tanh ^{-1}\left (\sqrt{1-x^3}\right )-\frac{2 \sqrt{2+\sqrt{3}} f (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*ArcTanh[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 = 12.2907, size = 114, normalized size = 0.85 \[ - \frac{2 e \operatorname{atanh}{\left (\sqrt{- x^{3} + 1} \right )}}{3} - \frac{2 \cdot 3^{\frac{3}{4}} f \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} \]

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*atanh(sqrt(-x**3 + 1))/3 - 2*3**(3/4)*f*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(
3))**2)*sqrt(sqrt(3) + 2)*(-x + 1)*elliptic_f(asin((-x - sqrt(3) + 1)/(-x + 1 +
sqrt(3))), -7 - 4*sqrt(3))/(3*sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 +
1))

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Mathematica [A]  time = 0.602799, size = 140, normalized size = 1.04 \[ \frac{2 f \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \sqrt{1-x^3}}-\frac{2}{3} e \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*ArcTanh[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 = 122, normalized size = 0.9 \[{-{\frac{2\,i}{3}}f\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}-{\frac{2\,e}{3}{\it Artanh} \left ( \sqrt{-{x}^{3}+1} \right ) } \]

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*arctanh((-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 = 5.58772, size = 65, normalized size = 0.49 \[ e \left (\begin{cases} - \frac{2 \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\\frac{2 i \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{otherwise} \end{cases}\right ) + \frac{f x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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