While I agree with MrHen in the general case, I do feel that this particular question has merit. The answer comes down to the order in which these words evolved, and the nature of their Latin roots. Language is not an exact science (far, far from it).
The short answer is that the word exply (or, rather, the meaning that it would convey) already exists in the word explain.
The "im" words in your list -- imply, implicate and implicit -- all evolved from the Latin implicare (im- + plicare) which means "to in-fold" or "involve". The word explicit has similar roots: it is based on the Latin explicitus (meaning "disentangled" or "easy"), past participle of explicare (ex- + plicare) which means "to out-fold or "unfold", making it the reverse of implicare.
The word explain, however, has its roots in the Latin explanare (ex- + planare), which means "to flatten out", or "make level".
Based on these origins, there was really only one choice for the concept of implicate, since it literally meant to fold in on itself. For the opposing concept, there were two choices: explicate (to unfold), or explain (to make flat). Note that these words are still valid synonyms in English. From this position it seems obvious why the word exply never appeared: at the time when imply was evolving from its Latin origins in implicate, the word explain already existed.
To summarize, picture the words evolving in parallel (from left to right) as follows:
plicare + in -> implicare -> implicate -> imply
+ ex -> explicare -> explicate
planare + ex -> explanare -> explain
Given that exply would have the same meaning as explain, there was simply no need for it to enter into the language.
Edit (in response to the modified question, and some comments):
Is explain the equivalent? It has the same Latin root, but doesn't have the full connotation of intentionally making something explicit.
I would argue that the word explain does imply a direct intention to make something explicit. A person seldom (if ever) explains something without intending to make it very clear to their audience. Also, it does have a different Latin root. See above!