(* The triangle has three vertex points (x0,y0) (x1,y1) (x2,y2).
We will integrate (x^alpha*y^beta) symbolicly on this triangle, where alpha and beta are positive integers.
lambda1 and lambda2 are the variables on the reference triangle.
*)

RegionPlot[lambda1+lambda2≤1,{lambda1,0,1},{lambda2,0,1}]

Clear[alpha,beta]

alpha = 1
beta = 1

Simplify[Integrate[ ((x1-x0)lambda1+(x2-x0)lambda2+x0)^alpha((y1-y0)lambda1+(y2-y0)lambda2+y0)^beta (Det[{{x1-x0,x2-x0},{y1-y0,y2-y0}}])
Boole[lambda1≤ 1 &&lambda1>=0&&lambda2≥  0&&lambda2≤  1&&lambda1+lambda2≤1], {lambda1,0,1},{lambda2,0,1} ] ]