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

Optimal. Leaf size=148 \[ \frac{4}{9} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right )-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.258307, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{4}{9} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right )-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 85.6724, size = 376, normalized size = 2.54 \[ \frac{2 \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \frac{\sqrt{3}}{3} + 1\right ) \left (- x + 1\right ) \operatorname{atan}{\left (\frac{3^{\frac{3}{4}} \sqrt{\sqrt{3} + 2} \sqrt{- \frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 1}}{3 \sqrt{\frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 4 \sqrt{3} + 7}} \right )}}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \sqrt{x^{3} - 1}} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 1\right ) \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \sqrt{x^{3} - 1}} + \frac{8 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (- 4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{4 \sqrt{3} + 7} \sqrt{x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**2)*(-sqrt(3)/3 + 1)*(-x + 1)*atan(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 - sqrt(3) + 1)**2)*(-sqrt(3) + 3)*sqrt(x**3 - 1)) - 2*3**(3/4)*sqrt((x**2 +
x + 1)/(-x - sqrt(3) + 1)**2)*(-sqrt(3) + 1)*sqrt(-sqrt(3) + 2)*(-x + 1)*ellipt
ic_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)) + 8*3**(1/4)*sqrt((x**2
+ x + 1)/(-x - sqrt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(-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))

_______________________________________________________________________________________

Mathematica [C]  time = 0.434571, size = 193, normalized size = 1.3 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (\frac{\left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{2 i \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1}-2}\right )}{\sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*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))] + ((2*I)*Sqrt[1 + x
+ x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1
+ (-1)^(1/3))]], (-1)^(1/3)])/(-2 + (-1)^(1/3))))/Sqrt[-1 + x^3]

_______________________________________________________________________________________

Maple [B]  time = 0.008, size = 240, normalized size = 1.6 \[ 2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }-{\frac{-6-2\,i\sqrt{3}}{3}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},{\frac{i}{6}}\sqrt{3}+{\frac{1}{2}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(2+x)/(x^3-1)^(1/2),x)

[Out]

2*(-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))-4/3*(-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),1/6*I*3^(1/2)+1/2,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(sqrt((x - 1)*(x**2 + x + 1))*(x + 2)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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