I have an integral, with $a,b >0$ :
$\int_0^\infty e^{-a^2 x^2 (1-x/b)^2}dx$
It's not diverging. Now let's change the variable : $z\rightarrow a x (1-x/b)$. The integral becomes :
$\int_{-\infty}^{z(ba/2)} e^{-z^2}\frac{b}{\sqrt{a} \sqrt{a b^2 - 4 z}}dz$
which has a singularity in $z=ab^2/4$. How did that happen ?
Your change of variable is not one-to-one, which invalidates the RHS. But, the existence of a singularity is not precluded by a change of variable, e.g. $$\int_0^\infty e^{-x^2/2} dx = \int_0^\infty e^{-y} (2y)^{-1/2} dy$$ because the integrand is still (Riemann) integrable.