LMS with CiviCRM/drupal integration?
It's difficult to answer your question without more details about what KIND of integration you're looking for, but I can give two answers:


If you're looking for a Learning Record Store (LRS), a native Drupal LRS exists, and CiviCRM integration for it exists too.
Some integration with Moodle exists via CiviMoodle - you can register someone for a CiviCRM event, and they're signed up for one or more corresponding events  in Moodle (with user creation if necessary).  
There's also some custom integration for pulling grades back into CiviCRM from Moodle, but would need some love from a developer to fit someone's situation.