It can be reduced to [Mordell equation](https://en.wikipedia.org/wiki/Mordell_curve):
$$Y^2 = X^3 + (4ABC)^2$$
with $Y:=4AB(2By^3-C)$ and $X:=-4ABxy$, which was shown by Mordell to have finitely many integer solutions.

**ADDED.** [M. A. Bennett and A. Ghadermarzi (2015)](http://www.math.ubc.ca/~bennett/BeGh-LMSJCM-2015.pdf) explored the connection between Mordell equations and cubic Thue equations, and computed all solutions for the former with the free terms below $10^7$ by absolute value.