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

Optimal. Leaf size=176 \[ \frac{2 \left (3-2 \sqrt [3]{2}\right ) \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{2 \left (2+3\ 2^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{3 \sqrt{3}} \]

[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 = 0.360584, antiderivative size = 176, 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 \[ \frac{2 \left (3-2 \sqrt [3]{2}\right ) \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{2 \left (2+3\ 2^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{3 \sqrt{3}} \]

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 = 148.138, size = 452, normalized size = 2.57 \[ - \frac{\sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (2 + 3 \cdot 2^{\frac{2}{3}}\right ) \left (- x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{\sqrt{3} + 2} \sqrt{- \frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 1}}{3 \sqrt{-1 + \sqrt [3]{2}} \sqrt{\frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 4 \sqrt{3} + 7}} \right )}}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{x^{3} - 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- 3 \sqrt{3} + 5\right ) \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1} \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} + \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (2 + 3 \cdot 2^{\frac{2}{3}}\right ) \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (\frac{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{4 \sqrt{3} + 7} \sqrt{x^{3} - 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} \]

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)*(2 + 3*2**(2/3))*(-x + 1)*atanh(3**(
3/4)*sqrt(1 + 2**(1/3))*sqrt(sqrt(3) + 2)*sqrt(-(-x + 1 + sqrt(3))**2/(x - 1 + s
qrt(3))**2 + 1)/(3*sqrt(-1 + 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(-1 + 2**(1/3
))*(1 + 2**(1/3))**(3/2)*sqrt(x**3 - 1)) + 2*3**(3/4)*sqrt((x**2 + x + 1)/(-x -
sqrt(3) + 1)**2)*(-3*sqrt(3) + 5)*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)*(2 + 3*2**(2/3))*sqrt(sqrt(3) + 2)*(-x + 1)*ellip
tic_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)*sqrt(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 = 0.658621, size = 333, normalized size = 1.89 \[ \frac{2 \sqrt [6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left (4 \sqrt{3} \left (3+\sqrt [3]{2}\right ) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )-3 i \sqrt{-2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x-\sqrt [3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt [3]{2}-6 i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (i+2 i 2^{2/3}+\sqrt{3}\right ) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^3-1}} \]

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*I)*Sqrt[-I + Sqrt[3] - (2*
I)*x]*(-6*I - (3*I)*2^(1/3) + 2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*
Sqrt[3] + 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 + S
qrt[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 + Sqrt[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 = 0.033, size = 266, normalized size = 1.5 \[ 2\,{\frac{ \left ( -2-3\,{2}^{2/3} \right ) \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1} \left ( -{2}^{2/3}+1 \right ) }\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},{\frac{3/2+i/2\sqrt{3}}{-{2}^{2/3}+1}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }-6\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \]

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

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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. \[ - \int \frac{3 x}{x \sqrt{x^{3} - 1} - 2^{\frac{2}{3}} \sqrt{x^{3} - 1}}\, dx - \int \frac{2}{x \sqrt{x^{3} - 1} - 2^{\frac{2}{3}} \sqrt{x^{3} - 1}}\, dx \]

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