If y=y(x) is the solution of the differential equation (d y)/(d x)+(4 x)/((x^2-1)) y=(x+2)/((x^2-1)^(5/2)), x>1 such that y(2)=2/9 log_e(2+√ 3 ) and…

If $\displaystyle y=y(x)$ is the solution of the differential equation $\displaystyle \frac{d y}{d x}+\frac{4 x}{\left(x^2-1\right)} y=\frac{x+2}{\left(x^2-1\right)^{\frac{5}{2}}}, x>1$ such that $\displaystyle y(2)=\frac{2}{9} \log _e(2+\sqrt{3})$ and $\displaystyle y(\sqrt{2})=\alpha \log _e(\sqrt{\alpha}+\beta)+\beta-\sqrt{\gamma}, \alpha, \beta, \gamma \in \mathbb{N}$, then $\displaystyle \alpha \beta \gamma$ is equal to $\displaystyle \_\_\_\_$