This can be entered into CVX as
-2*rel_entr(x+y,x+2*y) - rel_entr(x+2*y,x+y)
Derivation:
x*log(1+y/(x+y)) = (x+y)*log(1+y/(x+y)) - y*log(1+y/(x+y))
The first term is -rel_entr(x+y,x+2*y).
The second term can be handled with the result from the wizard of conic reformulation @Michal_Adamaszek in Writing x*log(1+x/y) , that
x*log(1+x/y) = rel_entr(x+y,y) + rel_entr(y,x+y) , which in this case becomes
-(rel_entr(x+2*y,x+y) + rel_entr(x+y,x+2*y)))