DruBuc,
For a long time my only form of transport was a Vespa Scooter, then a CBR 600 so well used to carrying rods on a bike
Firstly i can tell you what NOT to do...and that is try to ride with a beach caster slung over your shoulder! I lost the tip somewhere between Auchenblae and Stonehaven!
I have managed 3 x 12' pike rods, but its not pretty and it was only for a short distance.
I managed ok with a 4 piece rod (lots more to chose from on the market) in a tube. I either placed it into a rucsac on my back, or bungeed it to the seat so it ran along the seat..if that makes sense? If you have a trials style bike I have seen people fix a tube to the rear of the bike pointing up the way, then slide the rod tube into it. If you have a tourer with panniers you can bungee it across the seat and the top of the panniers
A 4 piece 9 footer works out about 30" is you include the tube, maybe a bit less, or 27 inches without a tube.
Plenty of 4 piece rods in the Orvis Sale in Banchory
Hope that helps a little.