The sum
$\sum_{a,b,c,d,e,f=1\wedge !(a=b \wedge c=d \wedge e=f)}^{n} (e-f)^2/((a-b)^2+(c-d)^2+(e-f)^2)^{3/2}$
was presented with a desire to evaluate it for $n$ up to 100. Clearly doing this directly becomes preposterous for large $n$, with the number of terms ~10^12 for $n=100$.
However, it can be observed that there are $n$ distinct possible values for $v_x=(e-f)^{2}$ and ~$2 n^{2}$ distinct possible values for $v_y=(a-b)^{2}+(c-d)^{2}$ up to multiplicities for each.
This then allows one to simply evaluate the combinations of $v_x$ and $v_y$, multiplying by the multiplicities of each, and adding that result to the total. This reduces the combinatorial explosion as $n$ increases significantly.
However, by $n\sim 1000$ even this method nears its breaking point.
Is there a more effective way of evaluating this sum for larger $n$, or an accurate approximation for $n>1000$?