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

Optimal. Leaf size=821 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}-\frac{\sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2^{5/6} \sqrt [4]{3} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{\log \left (\frac{(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{6 \sqrt [3]{2}}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac{\sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2} \]

[Out]

-((1 - x)^(2/3)*(2 - x)^(2/3))/(4*x^2) - ((1 - x)^(2/3)*(2 - x)^(2/3))/(2*x) - (
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))) - Ar
cTan[1/Sqrt[3] + (2^(1/3)*(2 - x)^(2/3))/(Sqrt[3]*(1 - x)^(1/3))]/(2*2^(1/3)*Sqr
t[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]*Elli
pticE[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 + Sqr
t[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2]) - (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^(1/4)*(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]) + Log[-(1 - x)^(1/3) +
(2 - x)^(2/3)/2^(2/3)]/(4*2^(1/3)) - Log[x]/(6*2^(1/3))

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Rubi [A]  time = 1.2221, antiderivative size = 821, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}-\frac{\sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2^{5/6} \sqrt [4]{3} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{\log \left (\frac{(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{6 \sqrt [3]{2}}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac{\sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((1 - x)^(2/3)*(2 - x)^(2/3))/(4*x^2) - ((1 - x)^(2/3)*(2 - x)^(2/3))/(2*x) - (
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))) - Ar
cTan[1/Sqrt[3] + (2^(1/3)*(2 - x)^(2/3))/(Sqrt[3]*(1 - x)^(1/3))]/(2*2^(1/3)*Sqr
t[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]*Elli
pticE[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 + Sqr
t[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2]) - (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^(1/4)*(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]) + Log[-(1 - x)^(1/3) +
(2 - x)^(2/3)/2^(2/3)]/(4*2^(1/3)) - Log[x]/(6*2^(1/3))

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Rubi in Sympy [A]  time = 43.5908, size = 731, normalized size = 0.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(2/3)*log(x)/12 + 2**(2/3)*log(-(-x + 1)**(1/3) + 2**(1/3)*(-x + 2)**(2/3)/2
)/8 - 2**(2/3)*sqrt(3)*atan(sqrt(3)/3 + 2**(1/3)*sqrt(3)*(-x + 2)**(2/3)/(3*(-x
+ 1)**(1/3)))/12 - 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))) + 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)) - 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))/(6*sqr
t((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)
) - (-x + 1)**(2/3)*(-x + 2)**(2/3)/(2*x) - (-x + 1)**(2/3)*(-x + 2)**(2/3)/(4*x
**2)

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Mathematica [C]  time = 0.350429, size = 225, normalized size = 0.27 \[ \frac{(1-x)^{2/3} \left (\frac{75 x F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x-1,1-x\right )}{(x-1) \left (3 F_1\left (\frac{5}{3};\frac{1}{3},2;\frac{8}{3};x-1,1-x\right )-F_1\left (\frac{5}{3};\frac{4}{3},1;\frac{8}{3};x-1,1-x\right )\right )-5 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x-1,1-x\right )}+\frac{16 (x-1) x F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x-1,1-x\right )}{(x-1) \left (3 F_1\left (\frac{8}{3};\frac{1}{3},2;\frac{11}{3};x-1,1-x\right )-F_1\left (\frac{8}{3};\frac{4}{3},1;\frac{11}{3};x-1,1-x\right )\right )-8 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x-1,1-x\right )}+5 (x-2) (2 x+1)\right )}{20 \sqrt [3]{2-x} x^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

((1 - x)^(2/3)*(5*(-2 + x)*(1 + 2*x) + (75*x*AppellF1[2/3, 1/3, 1, 5/3, -1 + x,
1 - x])/(-5*AppellF1[2/3, 1/3, 1, 5/3, -1 + x, 1 - x] + (-1 + x)*(3*AppellF1[5/3
, 1/3, 2, 8/3, -1 + x, 1 - x] - AppellF1[5/3, 4/3, 1, 8/3, -1 + x, 1 - x])) + (1
6*(-1 + x)*x*AppellF1[5/3, 1/3, 1, 8/3, -1 + x, 1 - x])/(-8*AppellF1[5/3, 1/3, 1
, 8/3, -1 + x, 1 - x] + (-1 + x)*(3*AppellF1[8/3, 1/3, 2, 11/3, -1 + x, 1 - x] -
AppellF1[8/3, 4/3, 1, 11/3, -1 + x, 1 - x]))))/(20*(2 - x)^(1/3)*x^2)

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Maple [F]  time = 0.115, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{2-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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