The pilot reported that he was asked by air traffic control to enter a right base, crossing in front of a passenger jet on final approach to another runway. The pilot further stated that he accepted this option and did not feel rushed; however, in doing so, it interrupted him from his typical traffic pattern entry and checklist usage, which resulted in him not extending the landing gear. The airplane subsequently landed gear up which resulted in substantial damage to the fuselage. The pilot reported no preaccident mechanical malfunctions or failures with the airplane that would have precluded normal operation.