For forms of the form $Ax^d - By^d$ this seems to have been done by Bennett, Dummigan, and Wooley: 

<cite cite="_Compositio Math._ **111** (1998), no. 1, 15--33" mrnumber="1611071" authors="M. A. Bennett, N. P. Dummigan, and T. D. Wooley">_M. A. Bennett, N. P. Dummigan, and T. D. Wooley_, [**The representation of integers by binary additive forms**](http://dx.doi.org/10.1023/A:1000270823971), _Compositio Math._ **111** (1998), no. 1, 15--33.</cite>