Correct syntax should be:
LineIntegrate[{2 x y, x^2 + y^2}, {x, y} \[Element]
ImplicitRegion[{x^2/25 + y^2/16 == 1, 0 <= x <= 5, 0 <= y <= 4}, {x,
y}]]
but I have to disappoint you ImplicitRegion dosen't work with LineIntegrate yet.
Maybe in future version of MMA can be implemented.
Maybe this helps.
Regards M.I.