3.867 \(\int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\)

Optimal. Leaf size=671 \[ -\frac{3 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{\sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{\sqrt [6]{2} 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{(3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}} \]

[Out]

(-3*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3))/(2^(1/3)*(3 - 2*
x)*(1 - x)^(1/3)*(2 - x)^(1/3)*(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))) +
(3*3^(1/4)*Sqrt[2 - Sqrt[3]]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3)*(1 + 2^(2/
3)*(2 - 3*x + x^2)^(1/3))*Sqrt[(1 - 2^(2/3)*(2 - 3*x + x^2)^(1/3) + 2*2^(1/3)*(2
- 3*x + x^2)^(2/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2]*EllipticE[
ArcSin[(1 - Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(1 + Sqrt[3] + 2^(2/3)*(2 -
3*x + x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2*2^(1/3)*(3 - 2*x)*Sqrt[(3 - 2*x)^2]*(1
- x)^(1/3)*(2 - x)^(1/3)*Sqrt[(1 + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(1 + Sqrt[3] +
2^(2/3)*(2 - 3*x + x^2)^(1/3))^2]) - (2^(1/6)*3^(3/4)*Sqrt[(-3 + 2*x)^2]*(2 - 3
*x + x^2)^(1/3)*(1 + 2^(2/3)*(2 - 3*x + x^2)^(1/3))*Sqrt[(1 - 2^(2/3)*(2 - 3*x +
x^2)^(1/3) + 2*2^(1/3)*(2 - 3*x + x^2)^(2/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x +
x^2)^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(
1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))], -7 - 4*Sqrt[3]])/((3 - 2*x)*Sqrt[
(3 - 2*x)^2]*(1 - x)^(1/3)*(2 - x)^(1/3)*Sqrt[(1 + 2^(2/3)*(2 - 3*x + x^2)^(1/3)
)/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2])

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Rubi [A]  time = 0.599281, antiderivative size = 671, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{3 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{\sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{\sqrt [6]{2} 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{(3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3))/(2^(1/3)*(3 - 2*
x)*(1 - x)^(1/3)*(2 - x)^(1/3)*(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))) +
(3*3^(1/4)*Sqrt[2 - Sqrt[3]]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3)*(1 + 2^(2/
3)*(2 - 3*x + x^2)^(1/3))*Sqrt[(1 - 2^(2/3)*(2 - 3*x + x^2)^(1/3) + 2*2^(1/3)*(2
- 3*x + x^2)^(2/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2]*EllipticE[
ArcSin[(1 - Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(1 + Sqrt[3] + 2^(2/3)*(2 -
3*x + x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2*2^(1/3)*(3 - 2*x)*Sqrt[(3 - 2*x)^2]*(1
- x)^(1/3)*(2 - x)^(1/3)*Sqrt[(1 + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(1 + Sqrt[3] +
2^(2/3)*(2 - 3*x + x^2)^(1/3))^2]) - (2^(1/6)*3^(3/4)*Sqrt[(-3 + 2*x)^2]*(2 - 3
*x + x^2)^(1/3)*(1 + 2^(2/3)*(2 - 3*x + x^2)^(1/3))*Sqrt[(1 - 2^(2/3)*(2 - 3*x +
x^2)^(1/3) + 2*2^(1/3)*(2 - 3*x + x^2)^(2/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x +
x^2)^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(
1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))], -7 - 4*Sqrt[3]])/((3 - 2*x)*Sqrt[
(3 - 2*x)^2]*(1 - x)^(1/3)*(2 - x)^(1/3)*Sqrt[(1 + 2^(2/3)*(2 - 3*x + x^2)^(1/3)
)/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2])

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Rubi in Sympy [A]  time = 21.2315, size = 612, normalized size = 0.91 \[ - \frac{3 \cdot 2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} \sqrt{4 x^{2} - 12 x + 9} \sqrt{\left (2 x - 3\right )^{2}}}{2 \left (- 2 x + 3\right ) \sqrt [3]{- x + 1} \sqrt [3]{- x + 2} \left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}\right )} + \frac{3 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2 \sqrt [3]{2} \left (x^{2} - 3 x + 2\right )^{\frac{2}{3}} - 2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1}{\left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1\right ) \sqrt [3]{x^{2} - 3 x + 2} \sqrt{\left (2 x - 3\right )^{2}} E\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} - \sqrt{3} + 1}{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{4 \sqrt{\frac{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1}{\left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}\right )^{2}}} \left (- 2 x + 3\right ) \sqrt [3]{- x + 1} \sqrt [3]{- x + 2} \sqrt{4 x^{2} - 12 x + 9}} - \frac{\sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} \left (x^{2} - 3 x + 2\right )^{\frac{2}{3}} - 2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1}{\left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}\right )^{2}}} \left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1\right ) \sqrt [3]{x^{2} - 3 x + 2} \sqrt{\left (2 x - 3\right )^{2}} F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} - \sqrt{3} + 1}{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1}{\left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}\right )^{2}}} \left (- 2 x + 3\right ) \sqrt [3]{- x + 1} \sqrt [3]{- x + 2} \sqrt{4 x^{2} - 12 x + 9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*2**(2/3)*(x**2 - 3*x + 2)**(1/3)*sqrt(4*x**2 - 12*x + 9)*sqrt((2*x - 3)**2)/(
2*(-2*x + 3)*(-x + 1)**(1/3)*(-x + 2)**(1/3)*(2**(2/3)*(x**2 - 3*x + 2)**(1/3) +
1 + sqrt(3))) + 3*2**(2/3)*3**(1/4)*sqrt((2*2**(1/3)*(x**2 - 3*x + 2)**(2/3) -
2**(2/3)*(x**2 - 3*x + 2)**(1/3) + 1)/(2**(2/3)*(x**2 - 3*x + 2)**(1/3) + 1 + sq
rt(3))**2)*sqrt(-sqrt(3) + 2)*(2**(2/3)*(x**2 - 3*x + 2)**(1/3) + 1)*(x**2 - 3*x
+ 2)**(1/3)*sqrt((2*x - 3)**2)*elliptic_e(asin((2**(2/3)*(x**2 - 3*x + 2)**(1/3
) - sqrt(3) + 1)/(2**(2/3)*(x**2 - 3*x + 2)**(1/3) + 1 + sqrt(3))), -7 - 4*sqrt(
3))/(4*sqrt((2**(2/3)*(x**2 - 3*x + 2)**(1/3) + 1)/(2**(2/3)*(x**2 - 3*x + 2)**(
1/3) + 1 + sqrt(3))**2)*(-2*x + 3)*(-x + 1)**(1/3)*(-x + 2)**(1/3)*sqrt(4*x**2 -
12*x + 9)) - 2**(1/6)*3**(3/4)*sqrt((2*2**(1/3)*(x**2 - 3*x + 2)**(2/3) - 2**(2
/3)*(x**2 - 3*x + 2)**(1/3) + 1)/(2**(2/3)*(x**2 - 3*x + 2)**(1/3) + 1 + sqrt(3)
)**2)*(2**(2/3)*(x**2 - 3*x + 2)**(1/3) + 1)*(x**2 - 3*x + 2)**(1/3)*sqrt((2*x -
3)**2)*elliptic_f(asin((2**(2/3)*(x**2 - 3*x + 2)**(1/3) - sqrt(3) + 1)/(2**(2/
3)*(x**2 - 3*x + 2)**(1/3) + 1 + sqrt(3))), -7 - 4*sqrt(3))/(sqrt((2**(2/3)*(x**
2 - 3*x + 2)**(1/3) + 1)/(2**(2/3)*(x**2 - 3*x + 2)**(1/3) + 1 + sqrt(3))**2)*(-
2*x + 3)*(-x + 1)**(1/3)*(-x + 2)**(1/3)*sqrt(4*x**2 - 12*x + 9))

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Mathematica [C]  time = 0.014997, size = 26, normalized size = 0.04 \[ -\frac{3}{2} (1-x)^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x-1\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3*(1 - x)^(2/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -1 + x])/2

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{2-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/(1-x)^(1/3)/(2-x)^(1/3),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.03536, size = 41, normalized size = 0.06 \[ - \frac{\left (-1\right )^{\frac{2}{3}} \left (x - 1\right )^{\frac{2}{3}} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\left (x - 1\right ) e^{2 i \pi }} \right )}}{\Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-1)**(2/3)*(x - 1)**(2/3)*gamma(2/3)*hyper((1/3, 2/3), (5/3,), (x - 1)*exp_pol
ar(2*I*pi))/gamma(5/3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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