Conrad Hilton raged at police during his arrest at E.G. Daily's house -- unleashing homophobic and racist slurs on police.

TMZ obtained video of Conrad's arrest early Saturday morning outside E.G.'s home, and he accused one of the officers of touching his penis, then screams a homophobic epithet. He also screamed rape. 

Law enforcement sources say he also used the n-word. The person who shot the video bleeped that word.

We broke the story ... Conrad was arrested for grand theft auto after allegedly stealing Rick Salomon's Bentley and driving it to E.G.'s house, where his ex-gf, Hunter Daily Salomon, lives. You can see the car in the video.

He also screamed "I'm Conrad motherf***ing Hilton, don't you forget it."

As we've reported ... E.G. believes Conrad has mental health issues and desperately needs to get treatment.