I need to run service to my motor home from the main house 100 amp panel 500 ft away to my motor home which has a 50 amp panel. What gauge is sufficient to achieve this either above ground or direct burial whichever is both cost efficient and safe.

This will give you an idea for 1/0 Aluminum. This calculation is based in part on the amperage actually used. You should provide service for 50 amps regardless of current draw, but the actual voltage drop will vary with the amount of current you use. If you don't use the full 50 amps the voltage drop will be smaller. 2-5% voltage drop is generally accepted. Closer to 2% is better. You could go a larger wire size, but I think you would be fine in this area. This is based on 240 volts at source. If your voltage is lower at the source, then the end voltage will be lower. I've included a link to a calculator to use. Plug in the numbers at 240 volts I am assuming and calculate.

Voltage Drop Calculator

Result

Voltage drop: 7.75 volts
Voltage drop percentage: 3.23%(larger wire size = smaller drop
Voltage at the end: 232.25 volts
Try this link for Calculator.net

Voltage Drop Calculator (http://www.calculator.net/voltage-drop-calculator.html)

As soon as you said 500' you threw cost efficient out the window. That is a loooong way to run line voltage (120/240V).

IMO 5% is perfectly acceptable depending on the starting voltage. Some places it's nearer to 250V so it can withstand move VD.

I would also think 1/0AL is an optimal mix of cost and power efficient.