Question: Recognize specific algebraic forms

Hi,

Wondered if anyone could help with the query below.

Consider f(x,y) defined as:
f := proc (x, y) options operator, arrow; x*y/(x+y) end proc

Then f(A, B); becomes:
(A * B )/(A + B)

now consider the polynomial:(poly2)

poly2:=(A*B+A*X+B*X)*(Y+X)/((A+B)*X*(2*Y+X))

This polynomial is just the expansion of the polynomial below (lets call it poly1) which MAPLE does not recognize.

(A*B/(A+B)+X)/(X+Y*X/(Y+X))

Here you can see that A,B on top and X,Y on the bottom are clearly of the form f(x,y).

Is there a way you can get MAPLE to recognize certain algebraic forms such that the polynomial poly2 could be written either as poly1 (already shown above) or as poly3 below:

poly3:=(f(A, B)+X)/(X+f(Y, X))

I have tried using simplify in the following form but not much luck. It doesn't seem to recognize anything other than the obvious.

simplify(poly2, {A*B/(A+B) = F1}, tdeg(A, B))

(I am still a bit new to the MAPLE syntax and procedures so apologies if I have missed something obvious function that can do this.)

Thanks.