Closed form or approximation of $\sum\limits_{i=0}^{n-1}\sum\limits_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)} (i + 2x)(j +2x)$ 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)?

Using Maple, I get
$$ (n+1) H(n) (n+4x-2)(x-1/2) + 2 n (n+2) x + \frac{n^3-3n}{2} $$
where $$H(n) = \sum_{k=1}^n 1/k = \Psi(n+1) + \gamma$$
As $n \to \infty$, 
$$ H(n) \sim \ln(n) + \gamma + \dfrac{1}{2n} - \dfrac{1}{12n^2}
+ \dfrac{1}{120 n^4} - \dfrac{1}{252 n^6} + \dfrac{1}{240 n^8} + \ldots $$
