Bright red Bleeding per rectum
can be due to Piles(Haemorrhoids),Cancer,Fissure in ano
Because of Analgesics and anti inflammatory drug,there may be ulcer in the stomach which bleeds sometimes.this blood will travel entire Gastro intestinal tract and come through anus dark coloured.
You will not pass huge amount of blood with constipation
.There may be some associated conditon.
Please consult your Gastroenterologist and General surgeon
and get examined clinically and investigated.
Hope it comes out to be a minor problem.
Wish you good health