2400 miles. Three days. 800 miles per day. That's 12-14 hours each day.
Completely do-able. However, there won't be much time for sightseeing and sitdown meals.
The short way would have you take I-44 between OKC and STL, to get from I-40 to I-70... That would allow you the opportunity for some great twisties in The Ozarks.
I can personally say that Fayetteville AR to St Louis on all twisty backroads will take you 8-9 hours. I-44 cuts that time in half.
If you're running ahead of schedule at Tulsa, take US-412 East. At Siloam Springs, take AR-16 East. You'll stay on AR-16 to Clinton. At Clinton, take AR-9 North. At the State Line, cross into Missouri and grab MO-19 at Thayer. Follow MO-19 North all the way to Cuba. Then, get back on I-44 and continue on.
You will have had your fill of Ozarks twisties by the time you reconnect with I-44. It's 600 miles from Tulsa to St Louis on this route, and you will have had a full day of riding by the time you see St Louis.
Ozarks Google Map Link
It's just over 1,000 miles from Phoenix to Tulsa. On those western highways, that's a 15 or 16 hour ride. If you knocked that out the 1st day, got a hot shower and a night's rest, you could hit the Ozarks on day two. Phoenix to Tulsa Google Map Link
On Day three, St Louis to Philadelphia. You'll have 14 or 15 hours of I-70 droning for the final 900 miles. STL to Philly Google Map Link