Question: how can I use dsolve/numeric with a numerical integration in the D.E. system?

I am facing a problem dsolve/numeric, and I am attaching two files: dsolve-issue.mws and dsolve-issue2.mws the first one shows a simplified version of the problem, the second one is an actual example of one the DE systems I desperately need to solve. There are brief comments in the files describing the problem, but here is the summary:

lets say we have the following differential equation:

 diff(x(t),t)= Int( f(x(t),p), p=p1..p2);

where p1 and p2 are real numbers, and f(x,p) is a function that cannot be integrated analatically [ that's why I used Int() instead of int() ]. However, assume that maple can integrate the right side numerically with no problems. In principle, maple should be able to solve this differential equation numerically, but no matter how hard I try to rewrite it in a friendly way for maple, it fails to do it.

looking at older questions and posts, i found a thread that solves the opposite of my problem, where it was explained how to numerically integrate the result of the dsolve/numeric. Note that what I am trying to do here is the exact opposite, numerically integrate the terms in the differential equaiton before solving it numerically at each point. here is the link to that post.

http://www.mapleprimes.com/questions/37157-Numerical-Integration-In-Phase-Space

I wrote my own simple code in maple to numerically solve the system with a second order Runge-Kutta method, but it is extremely slow, and not accurate enough to say the least, but it wroks in general. I just wanted to see that maple in principle should be able to do it.

any help would be greatly appreciated.