Try starting it up in "Safe Mode" and see if the power drains fast and it takes a long time to start. You can't do most things in that mode, but it is good to test in. If it starts up faster in that mode, and doesn't drain as fast, then you have something that is on the phone that is the problem. Virus, malware, bad app, etc., that is taking up resources, which drains your battery, and is possibly auto starting when the phone starts.
If it is fine in "Safe Mode" then check the device for the things mentioned. If you can't figure find it, then you may have to "Hard Reset" the phone, but that is a last resort as it will wipe the entire phone and you would have to back up everything (except apps, they can't be backed up) before doing it...