Because the denominator must be positive, the objective function, and hence the optimization problem is convex, and can be readily formulated and solved using CVX or a similar convex optimization tool.
cvx_begin
variable x(n)
minimize(sum(inv_pos(c + d.* x)))
x >= 0
sum(x) == C
cvx_end
You change x >= 0 to x >= small_number if you prefer. Strict inequalities are treated by the solvers as if they are non-strict inequalities.
It does not help to restrict x to integer values. That would make the problem more computationally difficult to solve. Nevertheless, to do so, merely change variable x(n) to variable x(n) integer .