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

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

[Out]

-3/2*2^(2/3)*(x^2-3*x+2)^(1/3)*((3-2*x)^2)^(1/2)*((-3+2*x)^2)^(1/2)/(3-2*x)/(1-x)^(1/3)/(2-x)^(1/3)/(1+2^(2/3)
*(x^2-3*x+2)^(1/3)+3^(1/2))-2^(1/6)*3^(3/4)*(x^2-3*x+2)^(1/3)*(1+2^(2/3)*(x^2-3*x+2)^(1/3))*EllipticF((1+2^(2/
3)*(x^2-3*x+2)^(1/3)-3^(1/2))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2)),I*3^(1/2)+2*I)*((-3+2*x)^2)^(1/2)*((1-2^(2
/3)*(x^2-3*x+2)^(1/3)+2*2^(1/3)*(x^2-3*x+2)^(2/3))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2))^2)^(1/2)/(3-2*x)/(1-x
)^(1/3)/(2-x)^(1/3)/((3-2*x)^2)^(1/2)/((1+2^(2/3)*(x^2-3*x+2)^(1/3))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2))^2)^
(1/2)+3/4*3^(1/4)*(x^2-3*x+2)^(1/3)*(1+2^(2/3)*(x^2-3*x+2)^(1/3))*EllipticE((1+2^(2/3)*(x^2-3*x+2)^(1/3)-3^(1/
2))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2)),I*3^(1/2)+2*I)*((-3+2*x)^2)^(1/2)*(1/2*6^(1/2)-1/2*2^(1/2))*((1-2^(2
/3)*(x^2-3*x+2)^(1/3)+2*2^(1/3)*(x^2-3*x+2)^(2/3))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2))^2)^(1/2)*2^(2/3)/(3-2
*x)/(1-x)^(1/3)/(2-x)^(1/3)/((3-2*x)^2)^(1/2)/((1+2^(2/3)*(x^2-3*x+2)^(1/3))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1
/2))^2)^(1/2)

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Rubi [A]  time = 0.29, antiderivative size = 671, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {61, 623, 303, 218, 1877} \[ -\frac {3 \sqrt {(3-2 x)^2} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{\sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )}-\frac {\sqrt [6]{2} 3^{3/4} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt {3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{(3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt {3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}}} \]

Antiderivative was successfully verified.

[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])

Rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[((a + b*x)^m*(c + d*x)^m)/(a*c + (b*
c + a*d)*x + b*d*x^2)^m, Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c -
a*d, 0] && LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4] && AtomQ[b*c + a*d]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 303

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sq
rt[2]*s)/(Sqrt[2 + Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a +
b*x^3], x], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 623

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[(d*Sqrt[(b + 2*c*x)
^2])/(b + 2*c*x), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx &=\frac {\sqrt [3]{2-3 x+x^2} \int \frac {1}{\sqrt [3]{2-3 x+x^2}} \, dx}{\sqrt [3]{1-x} \sqrt [3]{2-x}}\\ &=\frac {\left (3 \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{\sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=\frac {\left (3 \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {3}+2^{2/3} x}{\sqrt {1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{2^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}+\frac {\left (3 \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{\sqrt [6]{2} \sqrt {2+\sqrt {3}} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=-\frac {3 \sqrt {(3-2 x)^2} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{\sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\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 {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}-\frac {\sqrt [6]{2} 3^{3/4} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt {3}\right )}{(3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}\\ \end {align*}

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Mathematica [C]  time = 0.01, 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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fricas [F]  time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x + 2\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}}}{x^{2} - 3 \, x + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-x +1\right )^{\frac {1}{3}} \left (-x +2\right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (1-x\right )}^{1/3}\,{\left (2-x\right )}^{1/3}} \,d x \]

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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sympy [A]  time = 1.33, 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_polar(2*I*pi))/gamma(5/3)

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