iPhone 5 stuck on apple logo then shuts down

Unless you have a lot of experience doing board level troubleshooting, you best bet at recovering data is to get the phone to an iPhone data recovery specialist. However, if you want to try something yourself first, I would suggest the following:

• Open your phone and remove the logic board (follow this guide)
• Inspect the logic board, especially around the connectors and look for corrosion.
• Inspect both sides of the board. Unfortunately, most of the board is covered in shields. That's usually where the damage is occurring.
• Put your board in a container with >90% isopropyl alcohol and let it sit for a while.
• Use a soft brush, like a toothbrush and lightly brush away any corrosion you see.
• Rinse in alcohol and repeat.
• Let it air dry for a day.
• Re-assemble and hope for the best.

You should also replace the battery if it has swollen. Resist the temptation to pop it to let the gas out. A compromised Li-ion battery is a fire hazard. If the device appears to power up but behaves erratically, then use a tool like 3uTools to flash the firmware as it may be corrupted.

A professional repair shop that does water damage repair may be able to recover your data because they have access to pro-level ultrasonic baths and specialized cleaners as well as the skills to troubleshoot your board. Many shops have a no data/no fee policy so you don't have to spend money to find out if the data is recoverable or not.