The accident happened on the Edinburgh bypass between Baberton and Dreghorn at about 17:00 on Monday.
Police said a woman on the bus was taken to hospital with a "serious leg injury". Her injury is not thought to be life-threatening.
Four other people who suffered minor injuries were also taken to hospital but were later released.
The eastbound lane of the road was closed following the accident causing lengthy tailbacks. It was re-opened at 18:45.
Five people have been injured following a crash between a bin lorry, a bus and two cars.