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.
You want to compare a budget mini-computer that has the performance of a 5yo desktop to a system dedicated to just the single process of mining?
You could buy a lottery ticket for that 75$. It probably has the same chance of running a profit as mining on a rpi.
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.