If \( y=y(x) \) is the solution of the differential equation \( \dfrac{dy}{dx}+\dfrac{4x}{(x^{2}-1)}y=\dfrac{x+2}{(x^{2}-1)^{5/2}},\, x>1 \) such that \( y(2)=\dfrac{2}{9}\log_{e}(2+\sqrt{3}) \) and \( y(\sqrt{2})=\alpha \log_{e}(\sqrt{\alpha}+\beta)+\beta-\sqrt{\gamma},\; \alpha,\beta,\gamma \in \mathbb{N} \), then \( \alpha\beta\gamma \) is equal to __________.
