Given the Diophantine equation $$ x^2(8x-3)=y^2z, $$ is there a way to efficiently count the number of solutions that satisfy $x+y+z\leq n$, where $n$ is a fixed given integer?

Also, for any fixed $x$, is it possible to count all such solutions $(x,y,z)$ without having to explicitly find all the divisors of $x^2(8x-3)$?

A hint or a reference (if this is, in fact, easy) would be quite helpful.

I asked this on MSE, but got no responses.