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

Optimal. Leaf size=695 \[ -\frac{9 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{3\ 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2^{5/6} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{9 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{3}{4} (1-x)^{2/3} (2-x)^{2/3} \]

[Out]

(3*(1 - x)^(2/3)*(2 - x)^(2/3))/4 - (9*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2*x)^2]*(2 -
3*x + x^2)^(1/3))/(2*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))) + (9*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]])/(4*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]) - (3*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]])/(2^(5/6)*(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 = 0.759017, antiderivative size = 695, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{9 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{3\ 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2^{5/6} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{9 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{3}{4} (1-x)^{2/3} (2-x)^{2/3} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*(1 - x)^(2/3)*(2 - x)^(2/3))/4 - (9*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2*x)^2]*(2 -
3*x + x^2)^(1/3))/(2*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))) + (9*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]])/(4*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]) - (3*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]])/(2^(5/6)*(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 = 25.5274, size = 632, normalized size = 0.91 \[ \frac{3 \left (- x + 1\right )^{\frac{2}{3}} \left (- x + 2\right )^{\frac{2}{3}}}{4} - \frac{9 \cdot 2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} \sqrt{4 x^{2} - 12 x + 9} \sqrt{\left (2 x - 3\right )^{2}}}{4 \left (- 2 x + 3\right ) \sqrt [3]{- x + 1} \sqrt [3]{- x + 2} \left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}\right )} + \frac{9 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2 \sqrt [3]{2} \left (x^{2} - 3 x + 2\right )^{\frac{2}{3}} - 2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1}{\left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1\right ) \sqrt [3]{x^{2} - 3 x + 2} \sqrt{\left (2 x - 3\right )^{2}} E\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} - \sqrt{3} + 1}{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{8 \sqrt{\frac{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1}{\left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}\right )^{2}}} \left (- 2 x + 3\right ) \sqrt [3]{- x + 1} \sqrt [3]{- x + 2} \sqrt{4 x^{2} - 12 x + 9}} - \frac{3 \sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} \left (x^{2} - 3 x + 2\right )^{\frac{2}{3}} - 2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1}{\left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}\right )^{2}}} \left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1\right ) \sqrt [3]{x^{2} - 3 x + 2} \sqrt{\left (2 x - 3\right )^{2}} F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} - \sqrt{3} + 1}{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{2 \sqrt{\frac{2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1}{\left (2^{\frac{2}{3}} \sqrt [3]{x^{2} - 3 x + 2} + 1 + \sqrt{3}\right )^{2}}} \left (- 2 x + 3\right ) \sqrt [3]{- x + 1} \sqrt [3]{- x + 2} \sqrt{4 x^{2} - 12 x + 9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(-x + 1)**(2/3)*(-x + 2)**(2/3)/4 - 9*2**(2/3)*(x**2 - 3*x + 2)**(1/3)*sqrt(4*
x**2 - 12*x + 9)*sqrt((2*x - 3)**2)/(4*(-2*x + 3)*(-x + 1)**(1/3)*(-x + 2)**(1/3
)*(2**(2/3)*(x**2 - 3*x + 2)**(1/3) + 1 + sqrt(3))) + 9*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 + sqrt(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))/(8*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)) - 3*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))/(2*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 = 0.0271099, size = 38, normalized size = 0.05 \[ \frac{3}{4} (1-x)^{2/3} \left ((2-x)^{2/3}-3 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*(1 - x)^(2/3)*((2 - x)^(2/3) - 3*Hypergeometric2F1[1/3, 2/3, 5/3, -1 + x]))/4

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Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{x{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{2-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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