I want to know if a raspberry pi can mine bitcoin without using an ASIC miner.
You certainly can compile a Bitcoin node on a Raspberry Pi, and it will act as a fullnode relaying transactions for the network. For mining, while it probably would work technically, the chances of you ever mining a bitcoin are so low that its probably a waste of time.
You can run other coins based on PoS on a Raspberry Pi though and make money that way, I personally use Raspberry Pis as my Gridcoin and SolarCoin staking nodes.
Mining is about trying as many possible block candidates as possible. Your miner's proficiency at this task mostly corresponds to the amount of SHA-256d operations it can calculate. Raspberry Pis are computationally extremely weak hardware and are not optimized to run this operation. ASICs in turn, are application specific integrated circuits that have the SHA-256d operation baked into the hardware on the chip.
Your Raspberry Pi has very little computational power and a comparably bad energy efficiency. Chances are that you could run it for centuries without getting any reward.