Let $X$ and $Y$ be based spaces such that their respective loop spaces $\Omega X$ and $\Omega Y$ are connected. In the first paragraph of this article by Dula and Katz, it is given that $H_*(\Omega(X\vee Y))=H_*(\Omega X) \coprod H_*(\Omega Y)$ where $H_*$ is homology with field coefficients and $\coprod$ is the coproduct of differential graded algebras.
I am not really sure what is the coproduct of differential graded algebras. Is there an explicit formula for this coproduct in terms of $H_*(\Omega X)$ and $H_*(\Omega Y)$? In particular, if I know the betti numbers $b_*(\Omega X):=\dim H_*(\Omega)$ and $b_*(\Omega Y)$, is there a formula for the betti numbers of the coproduct?
I would also be grateful for a reference with explicit constructions where I can learn more about this.