Замена
[m]\sqrt{\frac{x-2}{x-1}}=t[/m] ⇒ [m]x=\frac{t^2-2}{t^2-1}[/m]
[m]dx=\frac{2tdt}{(t^2-1)^2}[/m]
[m]\frac{1}{(x-1)(x-2)}=\frac{1}{(\frac{t^2-2}{t^2-1}-1)(\frac{t^2-2}{t^2-1}-2)}=\frac{(t^2-1)^2}{t^2}[/m]
Подставляем
[m]\int\frac{dx}{\sqrt{(x-1)^3(x-2)}}=\int2dt=2t+C=2\sqrt{\frac{x-2}{x-1}}+C[/m]