If you are doing this to color soap you might want to move this question or thread to the CP soap section and you will get more responses.
Beet root will not color soap because it is pH sensitive and does not give good results for the high pH of soap. Actually it does not give good results for dying fabric or eggs either....
Madder root has worked for me, but a lot of the root is required to obtain a rather faint or weak red, very pretty but unfortunately it fades with sunlight. Not cost effective for me, but I am growing some madder root in the garden because it is a beautiful shade of red/rose/peach. If you grow it the cost is not so relevant.
Infused sweet paprika will work for a brick red, or a peach in lower concentrations, but it accelerates trace in CP soap and a weaker infusion will also fade in sunlight.
Alkanet works great and I have not found any problem with it, other than having to pay attention to any yellowing agents in the soap (either base oils or other additives) because yellow and purple makes grey.
Good luck to you and I hope this helped some.