Regional CRJ 900 Captain reported smoke in the flight deck and cabin just after takeoff resulting in an air turnback.

Narrative

On flight ZZZ - ZZZ1; right at liftoff from the runway we noticed smoke and fumes in the cockpit. The smoke appeared quickly and seemed to be increasing at first but then thinned out after about a minute. We had not changed to departure frequency yet and I called the tower to tell them we needed to return for landing immediately. Tower gave us vectors and then switched us to arrival. While coordinating with ATC I was acting as pilot monitoring and helping to configure the aircraft. The Flight Attendants chimed us and this gave me the indication that the cabin was getting the smoke/fumes as well. When I had a spare moment; I called back to the Flight Attendants (FAs) to ascertain the amount of smoke/fumes in the cabin and heard that it was not getting any worse; this was also the case for the flight deck the fumes seemed to be clearing. I made a PA to the passengers to tell them we were returning to ZZZ and landing ASAP. I sent an ACARS to dispatch that we were returning to ZZZ with smoke in the cabin. We briefed the ILS to Runway XX; reprogrammed the FMS; sent for landing numbers and then were vectored to the ILS Runway XX flew the approach and landed without incident. Airport Rescue and Firefighting (ARFF) met the aircraft on the taxiway and indicated there were no external indications of smoke or fire. Once I heard this I continued the taxi to the gate.Cause: After maintenance met us at the gate they noted a fan failure on the Maintenance Diagnostic Computer (MDC) computer.