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

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

[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))) - ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - x)^(2/3))/(Sqrt[3]*(1 - x)^(1/3))]/(2*2^(1/3)*Sqrt[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]])/(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]) - (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))

________________________________________________________________________________________

Rubi [A]  time = 0.570246, 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, Rules used = {124, 151, 157, 61, 623, 303, 218, 1877, 123} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}-\frac{\sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2^{5/6} \sqrt [4]{3} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{\log \left (\frac{(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{6 \sqrt [3]{2}}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac{\sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2} \]

Antiderivative was successfully verified.

[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))) - ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - x)^(2/3))/(Sqrt[3]*(1 - x)^(1/3))]/(2*2^(1/3)*Sqrt[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]])/(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]) - (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))

Rule 124

Int[((a_.) + (b_.)*(x_))^(m_)/(((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> Simp[(b*(a
+ b*x)^(m + 1)*(c + d*x)^(2/3)*(e + f*x)^(2/3))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[f/(6*(m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[((a + b*x)^(m + 1)*(a*d*(3*m + 1) - 3*b*c*(3*m + 5) - 2*b*d*(3*m + 7)*x))/((c + d*x
)^(1/3)*(e + f*x)^(1/3)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0] && ILtQ[m,
-1]

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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 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 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 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 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]

Rule 123

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[
(b*(b*e - a*f))/(b*c - a*d)^2, 3]}, -Simp[Log[a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[(Sqrt[3]*ArcTan[1/Sqrt[3
] + (2*q*(c + d*x)^(2/3))/(Sqrt[3]*(e + f*x)^(1/3))])/(2*q*(b*c - a*d)), x] + Simp[(3*Log[q*(c + d*x)^(2/3) -
(e + f*x)^(1/3)])/(4*q*(b*c - a*d)), x])] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^3} \, dx &=-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}+\frac{1}{24} \int \frac{24-4 x}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx\\ &=-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac{1}{48} \int \frac{-16-8 x}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx\\ &=-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}+\frac{1}{6} \int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx+\frac{1}{3} \int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx\\ &=-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\log \left (-\sqrt [3]{1-x}+\frac{(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{6 \sqrt [3]{2}}+\frac{\sqrt [3]{2-3 x+x^2} \int \frac{1}{\sqrt [3]{2-3 x+x^2}} \, dx}{6 \sqrt [3]{1-x} \sqrt [3]{2-x}}\\ &=-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\log \left (-\sqrt [3]{1-x}+\frac{(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{6 \sqrt [3]{2}}+\frac{\left (\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 )}{2 \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\log \left (-\sqrt [3]{1-x}+\frac{(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{6 \sqrt [3]{2}}+\frac{\left (\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^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}+\frac{\left (\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 )}{2 \sqrt [6]{2} \sqrt{2+\sqrt{3}} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=-\frac{(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac{(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac{\sqrt{(3-2 x)^2} \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{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{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-x)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\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 )}{4 \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{(-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 )}{2^{5/6} \sqrt [4]{3} (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{\log \left (-\sqrt [3]{1-x}+\frac{(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac{\log (x)}{6 \sqrt [3]{2}}\\ \end{align*}

Mathematica [C]  time = 0.0571213, size = 84, normalized size = 0.1 \[ -\frac{(1-x)^{2/3} \left (15 x^2 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x-1,1-x\right )+2 (x-1) x^2 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x-1,1-x\right )+5 (2-x)^{2/3} (2 x+1)\right )}{20 x^2} \]

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)^(2/3)*(1 + 2*x) + 15*x^2*AppellF1[2/3, 1/3, 1, 5/3, -1 + x, 1 - x] + 2*(-1 + x)*x^2
*AppellF1[5/3, 1/3, 1, 8/3, -1 + x, 1 - x]))/(20*x^2)

________________________________________________________________________________________

Maple [F]  time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{2-x}}}}\, dx \end{align*}

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-x + 2\right )}^{\frac{2}{3}}{\left (-x + 1\right )}^{\frac{2}{3}}}{x^{5} - 3 \, x^{4} + 2 \, x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-x + 2)^(2/3)*(-x + 1)^(2/3)/(x^5 - 3*x^4 + 2*x^3), x)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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