$\text{ Number of positive integers for which }\frac{(x+3)(x-1)}{x^2(x-2{)}^3}\le 0 holds is$

$\frac{(x+3)(x-1)}{x^2(x-2{)}^3}\le 0$

Sign scheme of the expression is as shown in the following figure

From the sign scheme

$x\in (-\mathrm{\infty },-3]\cup [1,2)$