Suppose we have the term $$\min\big\{A^2, \frac 1{B^2}\big\}(1+BA)$$ with $A\in\mathbb N$ and $0\leqslant B \leqslant 1$.
What is the best bound I can get on this quantity in the form $O(A^k$)? Bounding trivially we have $O(A^3)$, but this is insufficient for my purposes. It feels like the minimum and the term $BA$ are acting against each other, so the bound should be better, but I don't know how to translate my intuition to an actual better bound.
For fixed $B$, $A^2$ will exceed $\frac 1{B^2}$ eventually and the expression will be about $\frac AB$, so it is $O(A)$