Captain reported abnormal pack smell and smoke in the cabin after takeoff. The flight crew made an air turn back and precautionary landing at departure airport.

Narrative

We departed the gate and taxied without issue. We started the RH engine approximately 4 minutes before takeoff. Shortly after; we smelled a strong odor that wasn't consistent to the normal pack heat smell. I removed my sunglasses briefly after we rotated and saw what appeared to be smoke coming from the cockpit; we immediately closed our gaspers and heard the [priority] chime from the Flight Attendants. They informed us that there was smoke in the cabin. We let ATC know; [requested priority handling] and of our intention to return to the field. We determined since our gaspers were closed we did not need to don our masks and proceeded radar vectors to Runway XXR. We let ATC know that we wanted the trucks standing by due to us not knowing where the origin of the smoke was coming from. We ran the numbers; determined it was an overweight landing but not the greatest of concerns. Upon touchdown we rolled off; parked the plane; and proceeded to be inspected by ARF (airport rescue and fire). They determined they could not see any external source of fire and we taxi to the gate and deplaned.I don't see anything that could have been done better with the situation. Everyone performed their jobs and we were able to get back to the field with no delay. I am not a mechanic; but the engine was washed the night prior and the maintenance personnel informed me that was most likely the root cause due to excess soap left in the engine because they had not blown it out properly.