Before doing any mobilizing exercises it makes sense to somewhat get the blood flowing for a few minutes. I like to do this by either going to the grocery store or doing some other household chores for at least five to ten minutes, but if I'm running late I will do ten burpees instead to give myself a jump start.
The rest of the warmup should take only around 5 minutes.
Goal is just to loosen things up a little. Ten of each per direction:
The goal with the legs is to go through some range of motion as well as activating the muscles in the lower legs, especially around the ankles and feet. This focus on the ankles and feet is important for forefoot or midfoot strikers like myself. If you're a heel striker, you can still do these but might want to add something extra for the knees.
Staying on one leg, without holding onto anything, 10 each:
Then switch legs.
Since I don't hold onto anything, I have to balance myself during dynamic motion, which activates the muscles in the foot of the grounded leg. For me, this is the main purpose of the leg raises, with hip mobilization and activation of glutes and abs being merely a nice bonus.
With that I'm ready to head out the door, where I'll do a brisk walk for 2-3 minutes before starting to run.