3.63 \(\int \frac{e+f x}{(2-x) \sqrt{-1-x^3}} \, dx\)

Optimal. Leaf size=150 \[ \frac{2}{9} (e+2 f) \tan ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{-x^3-1}}\right )+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} (e-f) F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[Out]

(2*(e + 2*f)*ArcTan[(1 + x)^2/(3*Sqrt[-1 - x^3])])/9 + (2*Sqrt[2 - Sqrt[3]]*(e -
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*3^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[
3] + x)^2)]*Sqrt[-1 - x^3])

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Rubi [A]  time = 0.310743, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2}{9} (e+2 f) \tan ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{-x^3-1}}\right )+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} (e-f) F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(e + 2*f)*ArcTan[(1 + x)^2/(3*Sqrt[-1 - x^3])])/9 + (2*Sqrt[2 - Sqrt[3]]*(e -
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*3^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[
3] + x)^2)]*Sqrt[-1 - x^3])

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Rubi in Sympy [A]  time = 65.8967, size = 398, normalized size = 2.65 \[ \frac{\sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \frac{\sqrt{3}}{3} + 1\right ) \left (e + 2 f\right ) \left (x + 1\right ) \operatorname{atan}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2}}{3 \sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \sqrt{- x^{3} - 1}} + \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (e + 2 f\right ) \left (x + 1\right ) \Pi \left (- 4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{- x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{4 \sqrt{3} + 7} \sqrt{- x^{3} - 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) \left (e - f + \sqrt{3} f\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \sqrt{- x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt((x**2 - x + 1)/(x - sqrt(3) + 1)**2)*(-sqrt(3)/3 + 1)*(e + 2*f)*(x + 1)*ata
n(3**(3/4)*sqrt(1 - (x + 1 + sqrt(3))**2/(-x - 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2
)/(3*sqrt(4*sqrt(3) + 7 + (x + 1 + sqrt(3))**2/(-x - 1 + sqrt(3))**2)))/(sqrt((-
x - 1)/(x - sqrt(3) + 1)**2)*(-sqrt(3) + 3)*sqrt(-x**3 - 1)) + 4*3**(1/4)*sqrt((
x**2 - x + 1)/(x - sqrt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(e + 2*f)*(x + 1)*elliptic
_pi(-4*sqrt(3) + 7, asin((x + 1 + sqrt(3))/(-x - 1 + sqrt(3))), -7 + 4*sqrt(3))/
(sqrt((-x - 1)/(x - sqrt(3) + 1)**2)*(-sqrt(3) + 3)*(sqrt(3) + 3)*sqrt(4*sqrt(3)
+ 7)*sqrt(-x**3 - 1)) + 2*3**(3/4)*sqrt((x**2 - x + 1)/(x - sqrt(3) + 1)**2)*sq
rt(-sqrt(3) + 2)*(x + 1)*(e - f + sqrt(3)*f)*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
(3) + 3)*sqrt(-x**3 - 1))

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Mathematica [C]  time = 0.393274, size = 275, normalized size = 1.83 \[ \frac{2 \sqrt{\frac{2}{3}} \sqrt{-\frac{i (x+1)}{\sqrt{3}-3 i}} \left (2 \sqrt{3} \sqrt{2 i x+\sqrt{3}-i} \sqrt{x^2-x+1} (e+2 f) \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )-3 i f \sqrt{-2 i x+\sqrt{3}+i} \left (\left (\sqrt{3}-i\right ) x-\sqrt{3}-i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left (\sqrt{3}+3 i\right ) \sqrt{2 i x+\sqrt{3}-i} \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*Sqrt[2/3]*Sqrt[((-I)*(1 + x))/(-3*I + Sqrt[3])]*((-3*I)*f*Sqrt[I + Sqrt[3] -
(2*I)*x]*(-I - Sqrt[3] + (-I + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] +
(2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] + 2*Sqrt[3]*(e + 2*f)
*Sqrt[-I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sq
rt[3]), ArcSin[Sqrt[-I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*
I + Sqrt[3])]))/((3*I + Sqrt[3])*Sqrt[-I + Sqrt[3] + (2*I)*x]*Sqrt[-1 - x^3])

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Maple [A]  time = 0.01, size = 246, normalized size = 1.6 \[{{\frac{2\,i}{3}}f\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}+{\frac{{\frac{2\,i}{3}} \left ( e+2\,f \right ) \sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)/(2-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*I*(e+2*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)/(
-3/2+1/2*I*3^(1/2))*EllipticPi(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)),(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. \[ -\int \frac{f x + e}{\sqrt{-x^{3} - 1}{\left (x - 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(f*x + e)/(sqrt(-x^3 - 1)*(x - 2)),x, algorithm="maxima")

[Out]

-integrate((f*x + e)/(sqrt(-x^3 - 1)*(x - 2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{f x + e}{\sqrt{-x^{3} - 1}{\left (x - 2\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(f*x + e)/(sqrt(-x^3 - 1)*(x - 2)),x, algorithm="fricas")

[Out]

integral(-(f*x + e)/(sqrt(-x^3 - 1)*(x - 2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{e}{x \sqrt{- x^{3} - 1} - 2 \sqrt{- x^{3} - 1}}\, dx - \int \frac{f x}{x \sqrt{- x^{3} - 1} - 2 \sqrt{- x^{3} - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(e/(x*sqrt(-x**3 - 1) - 2*sqrt(-x**3 - 1)), x) - Integral(f*x/(x*sqrt(-
x**3 - 1) - 2*sqrt(-x**3 - 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{f x + e}{\sqrt{-x^{3} - 1}{\left (x - 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(f*x + e)/(sqrt(-x^3 - 1)*(x - 2)),x, algorithm="giac")

[Out]

integrate(-(f*x + e)/(sqrt(-x^3 - 1)*(x - 2)), x)