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

Optimal. Leaf size=343 \[ -\frac{3 (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \tan ^{-1}\left (\frac{\sqrt{\frac{13}{2}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}}}{\sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}}}\right )}{\sqrt{26} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{2 \sqrt{14+8 \sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{12 \sqrt [4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \Pi \left (97-56 \sqrt{3};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{2-\sqrt{3}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[Out]

(-3*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*ArcTan[(Sqrt[13/2]*Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2])/Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]])/(Sqrt[26]*Sq
rt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[-1 - x^3]) - (2*Sqrt[14 + 8*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^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*
Sqrt[-1 - x^3]) - (12*3^(1/4)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*El
lipticPi[97 - 56*Sqrt[3], -ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*S
qrt[3]])/(Sqrt[2 - Sqrt[3]]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[-1 - x^3])

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Rubi [A]  time = 1.45368, antiderivative size = 343, 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 \[ -\frac{3 (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \tan ^{-1}\left (\frac{\sqrt{\frac{13}{2}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}}}{\sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}}}\right )}{\sqrt{26} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{2 \sqrt{2 \left (7+4 \sqrt{3}\right )} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{12 \sqrt [4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \Pi \left (97-56 \sqrt{3};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{2-\sqrt{3}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

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]*ArcTan[(Sqrt[13/2]*Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2])/Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]])/(Sqrt[26]*Sq
rt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[-1 - x^3]) - (2*Sqrt[2*(7 + 4*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^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2
)]*Sqrt[-1 - x^3]) - (12*3^(1/4)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]
*EllipticPi[97 - 56*Sqrt[3], -ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 -
4*Sqrt[3]])/(Sqrt[2 - Sqrt[3]]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[-1 - x^3])

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Rubi in Sympy [A]  time = 90.658, size = 384, normalized size = 1.12 \[ - \frac{3 \sqrt{26} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) \operatorname{atanh}{\left (\frac{\sqrt{26} \cdot 3^{\frac{3}{4}} \sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2}}{6 \sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{26 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 1\right ) \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (\sqrt{3} + 2\right ) \sqrt{- x^{3} - 1}} + \frac{12 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) \Pi \left (\frac{\left (\sqrt{3} + 2\right )^{2}}{\left (-2 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{- x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 2\right ) \sqrt{\sqrt{3} + 2} \sqrt{4 \sqrt{3} + 7} \sqrt{- x^{3} - 1}} \]

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(26)*sqrt((x**2 - x + 1)/(x - sqrt(3) + 1)**2)*(x + 1)*atanh(sqrt(26)*3**
(3/4)*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)))/(26*sqrt((-x
- 1)/(x - sqrt(3) + 1)**2)*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)*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) + 2)*sqrt(-x**3 - 1)) + 12*3**(1/4)*sqrt((x**2 - x + 1)/(x -
sqrt(3) + 1)**2)*(x + 1)*elliptic_pi((sqrt(3) + 2)**2/(-2 + sqrt(3))**2, asin((x
+ 1 + sqrt(3))/(-x - 1 + sqrt(3))), -7 + 4*sqrt(3))/(sqrt((-x - 1)/(x - sqrt(3)
+ 1)**2)*(-sqrt(3) + 2)*sqrt(sqrt(3) + 2)*sqrt(4*sqrt(3) + 7)*sqrt(-x**3 - 1))

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Mathematica [C]  time = 0.406004, size = 196, normalized size = 0.57 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{\left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{3 i \sqrt{x^2-x+1} \Pi \left (\frac{i \sqrt{3}}{3+\sqrt [3]{-1}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{3+\sqrt [3]{-1}}\right )}{\sqrt{-x^3-1}} \]

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[(I*Sqrt[3])/(3 + (-1)^(1/3)), 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 = 0.01, size = 240, normalized size = 0.7 \[{-{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}+{\frac{2\,i\sqrt{3}}{{\frac{7}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{7}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

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)/(7/2+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)
/(7/2+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)