Phone boots for 5 seconds to apple logo then turns off

This is a belated answer so hopefully it can be of use to folks who search for and find this question.

This kind of symptom could simply be a bad battery. When the battery is too weak, it can't sustain the power requirements of a full boot sequence, let alone staying on. You could try to replace the battery by following this guide.

If you don't want to buy a battery for nothing, then disconnect the old battery and measure the voltage it generates directly at the connector on the battery flex. A fully charge battery will measure ~4.2V while a depleted battery will be ~ 3V. However, you have to be careful as some batteries offer the proper voltage but cannot sustain the significant current flow required during bootup. When this happens, the battery voltage drops precipitously which then causes the phone to turn off.