In 1961 Davenport showed that $H$ large enough there is a constant $c > 0$ such that $$ \sum \lvert D(P) \rvert^{-1/2} < c H^2 $$ where the sum is taken over the irreducible polynomials of degree $3$ with integer coefficients and height bounded by $H$. Here $D(P)$ denotes the discriminant of the polynomial $P$.
In the introduction he mentions that the irreducibility condition is only a matter of convenience, but he doesn't explicitly state that this can be relaxed. Furthermore, as far as I can tell the results on which the proof is based are only stated for irreducible forms.
So, my question is: does the above estimate holds when the sum is taken over polynomials with $D(P) \neq 0$, instead of just irreducible polynomials?
[1] Davenport, H. (1961). A note on binary cubic forms. Mathematika, 8(1), 58-62. doi:10.1112/S0025579300002138
