If $$y = y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} + \frac{4x}{x^2-1}y = \frac{x+2}{(x^2-1)^{5/2}}$$, $$x \gt 1$$ such that $$y(2) = \frac{2}{9}\log_e 2 + \sqrt{3}$$ 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 _____.