Cops in Cook County Illinois busted Smith on multiple charges last week -- including a count of rape, sexual battery and two counts of criminal confinement ... and the narrative spelled out in the arrest affidavit is insane.

In the docs, obtained by TMZ, the unidentified teen told cops she was kidnapped out of her room back in July after going to visit family members in Indiana. She initially said she'd exchanged Snapchat info with someone selling a cat, and gave them the address of the home where she was staying.

The alleged victim told police, that evening ... a man ripped open her bedroom window screen, yanked her out of the room, threw her into his truck and drove to a motel while sexually assaulting her. In later interviews, she said she'd actually linked with the alleged attacker on a dating app, and voluntarily snuck out to meet him.

In any case, once she met him in the truck she says she "immediately had a bad feeling" ... and he allegedly forced her to perform oral sex while driving to the motel, slapping and choking her multiple times, urinating on her in the motel room and forcing oral sex again.

Once he was finished, the teen claims he drove her back to her home and allegedly told her he'd be back the following night to pick her up again. When they pulled into her neighborhood, cops say her family was out front looking for her ... and as soon as they saw the truck, they all gave chase.

The alleged victim was able to escape, but the witnesses got enough info on the vehicle to relay to cops ... who then pinned it to Smith, who had appeared on the MTV show back in 2015.