Mathematics Question from JEE Main 2023 · 13 April, Shift 2

If $y = y(x)$ is the solution of the differential equation $\frac{dy}{dx} + \frac{4x}{\left(x^2 - 1\right)}\,y = \frac{x + 2}{\left(x^2 - 1\right)^{5/2}}$, $x > 1$, such that $y(2) = \frac{2}{9}\log_e\left(2 + \sqrt{3}\right)$ and $y\left(\sqrt{2}\right) = \alpha \log_e\left(\sqrt{\alpha} + \beta\right) + \beta - \sqrt{\gamma}$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha\beta\gamma$ is equal to _____.