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)