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

Optimal. Leaf size=727 \[ \frac{3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2-\frac{81 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{7 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{27 \sqrt [6]{2} 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{81 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{14 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{9}{70} (1-x)^{2/3} (2-x)^{2/3} (8 x+23) \]

[Out]

(3*(1 - x)^(2/3)*(2 - x)^(2/3)*x^2)/10 + (9*(1 - x)^(2/3)*(2 - x)^(2/3)*(23 + 8*
x))/70 - (81*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3))/(7*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))) + (81*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]])/(14*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]) - (27*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]])/
(7*(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 = 0.954787, antiderivative size = 727, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2-\frac{81 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{7 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{27 \sqrt [6]{2} 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{81 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{14 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{9}{70} (1-x)^{2/3} (2-x)^{2/3} (8 x+23) \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*(1 - x)^(2/3)*(2 - x)^(2/3)*x^2)/10 + (9*(1 - x)^(2/3)*(2 - x)^(2/3)*(23 + 8*
x))/70 - (81*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3))/(7*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))) + (81*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]])/(14*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]) - (27*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]])/
(7*(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 = 32.3269, size = 661, normalized size = 0.91 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**2*(-x + 1)**(2/3)*(-x + 2)**(2/3)/10 + 27*(-x + 1)**(2/3)*(-x + 2)**(2/3)*(
32*x/3 + 92/3)/280 - 81*2**(2/3)*(x**2 - 3*x + 2)**(1/3)*sqrt(4*x**2 - 12*x + 9)
*sqrt((2*x - 3)**2)/(14*(-2*x + 3)*(-x + 1)**(1/3)*(-x + 2)**(1/3)*(2**(2/3)*(x*
*2 - 3*x + 2)**(1/3) + 1 + sqrt(3))) + 81*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))/(28*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)) - 27*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*sq
rt(3))/(7*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 = 0.0477162, size = 49, normalized size = 0.07 \[ \frac{3}{70} (1-x)^{2/3} \left ((2-x)^{2/3} \left (7 x^2+24 x+69\right )-135 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*(1 - x)^(2/3)*((2 - x)^(2/3)*(69 + 24*x + 7*x^2) - 135*Hypergeometric2F1[1/3,
2/3, 5/3, -1 + x]))/70

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Maple [F]  time = 0.11, size = 0, normalized size = 0. \[ \int{{x}^{3}{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{2-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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