For forms of the form $Ax^d - By^d$ this seems to have been done by Bennett, Dummigan, and Wooley:
Bennett, M. A.(1-MI); Dummigan, N. P.(1-MI); Wooley, T. D.(1-MI)
The representation of integers by binary additive forms. (English summary)
Compositio Math. 111 (1998), no. 1, 15–33.
11D85 (11D41 11P05)