When is a probability density function square-integrable?

Consider a measure space $(X, mathcal{F}, mu)$ and let $f in L^1(X, mathcal{F}, mu)$ with $f(x) >0$ for all $x in X$ be a probability density function.
As discussed in this question, $f$ need not be in $L^2(X, mathcal{F}, mu)$. Moreover, $f$ can be continuous and differentiable and still not be square-integrable.

My question is if there are “simple” assumptions one can make about $f$ such that it lies in $L^2(X, mathcal{F}, mu)$.