During the solution of my programming problem I ended up with the following double sum:

$$\sum\_{i=0}^{n-1}\sum\_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)}\cdot (i + 2x)(j +2x)$$

where $x$ is some number. Because of the double sum the complexity of the problem will be quadratic (and I have $n$ at the scale of a million), but if I can find a closed form solution, it will reduce dramatically (log or even something close to constant).

After trying to simplify the sum using the fact that $ \frac{i + j + 2}{(i + 1)(j+1)} = \frac{1}{i+1} + \frac{1}{j+1}$ I think that my knowledge is not enough.

Can anyone help me to simplify this problem (or may be find a reasonable approximation)?