Посчитать = to count (number of students, apples, whatever)

Подсчитать = to count but using some effort (867 roubles + 300 roubles for delivery = ...)

Рассчитать = to calculate using mathematics, computer programms - the capacity in construction and so on
or (Рассчитать прислугу (old\rarely) = to pay wage to servants)