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

Optimal. Leaf size=145 \[ \frac{4}{9} \tanh ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{1-x^3}}\right )-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

[Out]

(4*ArcTanh[(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])

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Rubi [A]  time = 0.277236, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{4}{9} \tanh ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{1-x^3}}\right )-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(4*ArcTanh[(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])

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Rubi in Sympy [A]  time = 89.1562, size = 379, normalized size = 2.61 \[ \frac{2 \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (\frac{\sqrt{3}}{3} + 1\right ) \left (- x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 - \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2}}{3 \sqrt{- 4 \sqrt{3} + 7 + \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{- x^{3} + 1}} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{- x^{3} + 1}} - \frac{8 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{x - 1 + \sqrt{3}}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{- x^{3} + 1}} \]

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 + 1 + sqrt(3))**2)*(sqrt(3)/3 + 1)*(-x + 1)*atanh(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 + 1 + sqrt(3))**2)*(sqrt(3) + 3)*sqrt(-x**3 + 1)) - 2*3**(3/4)*sqrt((x**2
+ x + 1)/(-x + 1 + sqrt(3))**2)*(1 + sqrt(3))*sqrt(sqrt(3) + 2)*(-x + 1)*ellipt
ic_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)) - 8*3**(1/4)*sqrt((x**2
+ x + 1)/(-x + 1 + sqrt(3))**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 + 1 + sqrt(3))**2)*sqrt(-4*sqrt(3) + 7)*(-sqrt(3) + 3)*(sqrt(3) + 3)*sqrt
(-x**3 + 1))

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Mathematica [C]  time = 0.451624, size = 195, normalized size = 1.34 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (\frac{\left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{2 i \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1}-2}\right )}{\sqrt{1-x^3}} \]

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]

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Maple [A]  time = 0.009, size = 240, normalized size = 1.7 \[{-{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}+{\frac{{\frac{4\,i}{3}}\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(2+x)/(-x^3+1)^(1/2),x)

[Out]

-2/3*I*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))+4/3*I*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. \[ \int \frac{x}{\sqrt{-x^{3} + 1}{\left (x + 2\right )}}\,{d x} \]

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)

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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)

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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)

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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)