Can we get closed form solution for such a problem?

If some $A_i$ is negative the problem is unbounded: we can make the objective arbitrarily small by making $x_i$ arbitrarily close to 0.

Assuming $A_i geq 0$, the optimality is obtained when all $frac{A_i}{x_i^2}$ are equal (KKT optimality conditions), or equivalently all $frac{x_i^2}{A_i}$ are equal, and $sum x_i = X$ (otherwise you can increase some $x_i$ and reduce the objective).

Stating $x_i = sqrt{A_i} y$, you can deduce $y = frac{X}{sum sqrt{A_i}}$ from the equality.

Therefore $x_i = frac{sqrt{A_i}}{sum sqrt{A_j}}X$