A white Seat car crashed into a lamppost on Bradford Road, Northowram at about 8.50am on Monday, May 27.
It happened whilst the car was being pursued by a police car.
West Yorkshire Fire and Rescue Service said: "1 vehicle involved with 3 persons trapped.
"All three casualties extricated by Fire Service Personnel and handed over to the ambulance service."
Fire crews from Halifax and Illingworth attended.
A 31-year-old man was arrested on suspicion of driving whilst disqualified, aggravated vehicle taking and driving whilst over the limit.
A 26-year-old man was arrested on suspicion of driving whilst over the limit and aggravated vehicle taking.
He was taken to hospital for treatment for a serious arm injury.
A 29-year-old woman was arrested on suspicion of aggravated vehicle taking and driving whilst over the limit.
All three have been released while investigations continue.
Bradford Road was closed both ways because of the accident.
Buses in Halifax were diverted.
First Bus said in a statement on Twitter: "Due to RTC after Northowram Buses are diverted from shelf roundabout."
Metro Travel News said buses are unable to serve Northowram.
Anyone with information is asked to contact police on 101, quoting log 460 of 27/5.