you can do it in 9 hours - if you only stop for gas.....
with a child forget it.
It only takes me 12 hours when Spicey, my cat, is awake - he knows each and every rest stop - and will tell me about it if I miss even one....

all the way until the next rest stop.
but only when he is awake...
10 hours is generally enough if everyone else is sleeping - you get an half hour for meals, and another half hour for the rest stops....
I don't like going through Atlanta - my brother lives in Stockbridge, GA - so we go over there alot - I 20 can be awful, and I 285 has so many wrecks...
not to mention all the construction now - someone else says they have closed all but one lane on I75 in South Ga....
take I65 to I85 (montgomery) get off on exit 9 - turn right - when this roads deadends you will be on US 231. turn left.
I would stick on this route until you get to I10 in Florida - it has rest stops - traveling with a child you will need rest stops. You can get to I10 in around 4 to 3 1/2 hours from Birmingham.
don't turn off of US 231 until you get to I10.
my other short cut is at the end - get on the Florida turnpike - about 8 miles take the first REAL exit off - pay your $1.00 - then turn right - you are now on US 27. about 22 (28??) miles down this road will be a right turn (not left) to US 192. (on us 27 you will go through 2 small towns).
on US 192 (get in the middle lane) you will pass on the left - a publix - then some hotels, Orange Lake - then at 2 lights pass Orange Lake and another publix (get in the left lane) - is Black Lake road - the sign is above the traffic lights - turn left - this road will deadend at Sherbert Rd - turn left - at the next traffic light - turn right - you are now on WDW property (turning left at this light will be AKL) - follow the signs to the theme parks closest to your resort.
to come back this way look for the AKL (not ak).
of course I don't travel during busy times - I leave around midnight to 4 am in the morning - Mother once decided it was better to leave around 10 pm - bad, bad mistake - we got there around 5:30am - nothing at WDW was open....
I am better off driving at night - the Florida sun can be bad.
forgot buy gas in Dothan, then do what I learned here - find a stop with a bunch of gas stations -especially the cheap brands - stop there for gas - except to pay the most for gas along I10.
the turn off to I75 is about 2 miles past the rest stop (there are around 5 on I10) - then and only then will you start to see the signs for I75.