The Rav of our shul spoke about this when discussing our shul minhag of not saying tachanun until Rosh Chodesh Cheshvan. Note that this is our shul minhag as other shuls in the community start tachanun after Isru Chag (or maybe on Isru Chag, I am not sure).
The first part of the month continues tachanun because of the inyan of teshuvah and din. After Succos, we continue not saying tachanun because of the inyan of simcha and adding to Shmini Atzeres because it is a "Regel bifnei Atzmo" rather than the end of the chag (like the last day of Pesach). We also do not say tachanun for the week after Shavuos since there was a week afterwards that the korbon chagigah could be brought. Even though a separate chagigah was not brought on Shmini Atzeres (and the Ymei miluim of Succos are over) we still extend the simcha of the Chag.
This is from memory as to the explanation that our Rav gave for our shul minhag (which goes back to the founding of our shul as a separate minyon. Note that other shuls do not have this minhag (as I said earlier).
I do not recall the sources that our rav brought. Since there are those that do say tachanun, there is what to rely on in both ways.