Oblong numbers| primes solutions of p(p+1) + q(q+1) = r(r+1)

An Oblong number is a number which is the product of two consecutive integers, that is, a number of the form   $n \,(n + 1)$

Find primes solutions   $p, \; q, \; r$   of the equation

$p \,(p+1) \; + \; q \,(q+1) \; = \; r \,(r+1)$

Prove that the solution is unique.