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

Optimal. Leaf size=139 \[ \frac{2}{9} (e+2 f) \tanh ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{x^3+1}}\right )+\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (e-f) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

[Out]

(2*(e + 2*f)*ArcTanh[(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 = 0.277653, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2}{9} (e+2 f) \tanh ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{x^3+1}}\right )+\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (e-f) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(e + 2*f)*ArcTanh[(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 = 93.7997, size = 391, normalized size = 2.81 \[ \frac{\sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (\frac{\sqrt{3}}{3} + 1\right ) \left (e + 2 f\right ) \left (x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{- \sqrt{3} + 2} \sqrt{- \frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} + 1}}{3 \sqrt{\frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{x^{3} + 1}} - \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (e + 2 f\right ) \left (x + 1\right ) \Pi \left (4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{- x - 1 + \sqrt{3}}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{x^{3} + 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (e - f \left (1 + \sqrt{3}\right )\right ) \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{x^{3} + 1}} \]

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 + 1 + sqrt(3))**2)*(sqrt(3)/3 + 1)*(e + 2*f)*(x + 1)*atan
h(3**(3/4)*sqrt(-sqrt(3) + 2)*sqrt(-(-x - 1 + sqrt(3))**2/(x + 1 + sqrt(3))**2 +
1)/(3*sqrt((-x - 1 + sqrt(3))**2/(x + 1 + sqrt(3))**2 - 4*sqrt(3) + 7)))/(sqrt(
(x + 1)/(x + 1 + sqrt(3))**2)*(sqrt(3) + 3)*sqrt(x**3 + 1)) - 4*3**(1/4)*sqrt((x
**2 - x + 1)/(x + 1 + sqrt(3))**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 + 1 + sqrt(3))**2)*sqrt(-4*sqrt(3) + 7)*(-sqrt(3) + 3)*(sqrt(3)
+ 3)*sqrt(x**3 + 1)) + 2*3**(3/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*sqrt
(sqrt(3) + 2)*(e - f*(1 + sqrt(3)))*(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(3)
+ 3)*sqrt(x**3 + 1))

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Mathematica [C]  time = 0.405898, size = 273, normalized size = 1.96 \[ \frac{2 \sqrt{\frac{2}{3}} \sqrt{-\frac{i (x+1)}{\sqrt{3}-3 i}} \left (2 \sqrt{3} \sqrt{2 i x+\sqrt{3}-i} \sqrt{x^2-x+1} (e+2 f) \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )-3 i f \sqrt{-2 i x+\sqrt{3}+i} \left (\left (\sqrt{3}-i\right ) x-\sqrt{3}-i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left (\sqrt{3}+3 i\right ) \sqrt{2 i x+\sqrt{3}-i} \sqrt{x^3+1}} \]

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 [B]  time = 0.01, size = 246, normalized size = 1.8 \[ -2\,{\frac{f \left ( 3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }+{\frac{ \left ( 2\,e+4\,f \right ) \left ({\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }{3}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},{\frac{1}{2}}-{\frac{i}{6}}\sqrt{3},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \]

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*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+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)*EllipticPi(((1+x)/(3/2-1/2*I
*3^(1/2)))^(1/2),1/2-1/6*I*3^(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. \[ -\int \frac{f x + e}{\sqrt{x^{3} + 1}{\left (x - 2\right )}}\,{d x} \]

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)