The moon is accelerating due to the "sling" effect of the tides. Meaning, the center of mass of the ocean water pulled up by the gravity of the Moon itself is higher on that side than the center of mass of the earth. That difference provides enough centrifugal energy to the Moon to accelerate it.

However ...

If the Moon was actually part of the Earth from the beginning, according to this theory IT WOULD HAVE ALREADY LEFT ORBIT.