3.32 \(\int \frac{2+3 x}{\left (2^{2/3}+x\right ) \sqrt{-1-x^3}} \, dx\)

Optimal. Leaf size=169 \[ \frac{2 \left (2-3\ 2^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{-x^3-1}}\right )}{3 \sqrt{3}}+\frac{2 \left (3+2 \sqrt [3]{2}\right ) \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[Out]

(2*(2 - 3*2^(2/3))*ArcTanh[(Sqrt[3]*(1 + 2^(1/3)*x))/Sqrt[-1 - x^3]])/(3*Sqrt[3]
) + (2*(3 + 2*2^(1/3))*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 = 0.372021, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 \left (2-3\ 2^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{-x^3-1}}\right )}{3 \sqrt{3}}+\frac{2 \left (3+2 \sqrt [3]{2}\right ) \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(2 - 3*2^(2/3))*ArcTanh[(Sqrt[3]*(1 + 2^(1/3)*x))/Sqrt[-1 - x^3]])/(3*Sqrt[3]
) + (2*(3 + 2*2^(1/3))*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 = 144.832, size = 462, normalized size = 2.73 \[ \frac{\sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- 3 \cdot 2^{\frac{2}{3}} + 2\right ) \left (x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2}}{3 \sqrt{-1 + \sqrt [3]{2}} \sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{- x^{3} - 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- 3 \sqrt{3} + 1\right ) \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1} \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} - \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- 3 \cdot 2^{\frac{2}{3}} + 2\right ) \sqrt{\sqrt{3} + 2} \left (x + 1\right ) \Pi \left (\frac{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{- x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{4 \sqrt{3} + 7} \sqrt{- x^{3} - 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.677004, size = 338, normalized size = 2. \[ \frac{2 \sqrt [6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (3 \sqrt{2 i x+\sqrt{3}-i} \left (\left (3 \sqrt [3]{2}+4 i \sqrt{3}+i \sqrt [3]{2} \sqrt{3}\right ) x+i \sqrt [3]{2} \sqrt{3}-2 i \sqrt{3}-3 \sqrt [3]{2}-6\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )-4 \sqrt{3} \left (\sqrt [3]{2}-3\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (i+2 i 2^{2/3}+\sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*2^(1/6)*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(3*Sqrt[-I + Sqrt[3] + (2*I)*x]*(-6
- 3*2^(1/3) - (2*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3] + (3*2^(1/3) + (4*I)*Sqrt[3] +
I*2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1
/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] - 4*Sqrt[3]*(-3 + 2^(1/3))*Sqrt[I + Sqrt[3]
- (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]
), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqr
t[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt
[-1 - x^3])

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Maple [A]  time = 0.034, size = 253, normalized size = 1.5 \[{\frac{-{\frac{2\,i}{3}} \left ( 2-3\,{2}^{2/3} \right ) \sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{2}^{{\frac{2}{3}}}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{2}^{{\frac{2}{3}}}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}-{2\,i\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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