Actually they will, but the prices are pretty high. It looks like you have a choice between a flat fee of $67 or $41 an hour for the truck and crew.
http://abilenetx.gov/business/city-fee-schedule/fee-schedule#solidwaste
Looks like it would be cheaper to hire a Mexican with a pickup truck. Or at those prices, cut it up an throw a bag's worth in the garbage every week.
Apparently, several neighbors share a 300 gallon container, so if he filled it up with the mattress there was no place for the shooter to put his household garbage that week. (Curbside customers get individual 96 gallon rolling carts) http://abilenetx.gov/city-hall/departments/public-works/solid-waste-services/collection-services
Municipal garbage pickup, almost as evil as public school, here resulting in homicide. My private trash pickup will take anything they can shove in the truck, including rolled up carpet, with stuff inside.