Let $A, B, C, D \in \mathbb{R^*_+}$.

Is it possible to solve $$ \max_{ \substack{0 \leq x\leq A \\ 0\leq y\leq B}} \frac{1+x+y}{(1+Cx)(1+Dy)} $$

The KKT conditions give for an extrema $(x^*,y^*)$

If $\frac{1}{C} > (1+y^*)$ then $x^*=A$ otherwise $x^*=0$,

If $\frac{1}{D} > (1+x^*)$ then $y^*=B$ otherwise $y^*=0$.

Thus it solves the problem when $1/C>1+B$ and $1/D>1+A$.

But then I am stuck for the case $1/C\leq 1+B$ or $1/D\leq 1+A$