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

Optimal. Leaf size=379 \[ \frac{3 (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \tanh ^{-1}\left (\frac{\sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}}}{2 \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}}}\right )}{2 \sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{2 \sqrt{2 \left (37+20 \sqrt{3}\right )} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{13 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{12 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{1}{169} \left (553+304 \sqrt{3}\right );-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{13 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

[Out]

(3*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*ArcTanh[(Sqrt[7]*Sqrt[(1 - x)
/(1 + Sqrt[3] - x)^2])/(2*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2])])/(2*Sqrt[7]*
Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3]) - (2*Sqrt[2*(37 + 20*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]])/(13*3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] -
x)^2]*Sqrt[1 - x^3]) - (12*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/
(1 + Sqrt[3] - x)^2]*EllipticPi[(553 + 304*Sqrt[3])/169, -ArcSin[(1 - Sqrt[3] -
x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(13*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sq
rt[1 - x^3])

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Rubi [A]  time = 1.61058, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{3 (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \tanh ^{-1}\left (\frac{\sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}}}{2 \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}}}\right )}{2 \sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{2 \sqrt{2 \left (37+20 \sqrt{3}\right )} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{13 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{12 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{1}{169} \left (553+304 \sqrt{3}\right );-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{13 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*ArcTanh[(Sqrt[7]*Sqrt[(1 - x)
/(1 + Sqrt[3] - x)^2])/(2*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2])])/(2*Sqrt[7]*
Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3]) - (2*Sqrt[2*(37 + 20*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]])/(13*3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] -
x)^2]*Sqrt[1 - x^3]) - (12*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/
(1 + Sqrt[3] - x)^2]*EllipticPi[(553 + 304*Sqrt[3])/169, -ArcSin[(1 - Sqrt[3] -
x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(13*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sq
rt[1 - x^3])

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Rubi in Sympy [A]  time = 90.4734, size = 384, normalized size = 1.01 \[ \frac{3 \sqrt{7} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (- x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{7} \sqrt{1 - \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2}}{6 \sqrt{- 4 \sqrt{3} + 7 + \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}}} \right )}}{14 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 4\right ) \sqrt{- x^{3} + 1}} - \frac{12 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (\frac{\left (\sqrt{3} + 4\right )^{2}}{\left (-4 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x - 1 + \sqrt{3}}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \left (- \sqrt{3} + 4\right ) \left (\sqrt{3} + 4\right ) \sqrt{- x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(7)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*(-x + 1)*atanh(3**(3/4)*sqr
t(7)*sqrt(1 - (x - 1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)/(6*
sqrt(-4*sqrt(3) + 7 + (x - 1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2)))/(14*sqrt((-x
 + 1)/(-x + 1 + sqrt(3))**2)*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)*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) + 4)*sqrt(-x**3 + 1)) - 12*3**(1/4)*sqrt((x**2 + x + 1)/
(-x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(-x + 1)*elliptic_pi((sqrt(3) + 4)**2/
(-4 + sqrt(3))**2, 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) + 4)*(sqrt(3
) + 4)*sqrt(-x**3 + 1))

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Mathematica [C]  time = 0.399102, size = 195, normalized size = 0.51 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (\frac{\left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{3 i \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{5 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1}-3}\right )}{\sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/((3 + 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))] + ((3*I)*Sqrt[1 + x
+ x^2]*EllipticPi[(2*Sqrt[3])/(5*I + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1
 + (-1)^(1/3))]], (-1)^(1/3)])/(-3 + (-1)^(1/3))))/Sqrt[1 - x^3]

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Maple [A]  time = 0.011, size = 240, normalized size = 0.6 \[{-{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}+{\frac{2\,i\sqrt{3}}{{\frac{5}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{5}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(3+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))+2*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)/(5/2+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)/(5/2+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. \[ \int \frac{x}{\sqrt{-x^{3} + 1}{\left (x + 3\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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