Area of triangle

Find formula for the area of ​​the triangle, knowing its perimeter, the sum of the squares of the sides and the sum of cubes of the sides.

$\displaystyle a+b+c=p_1$ $a^2+b^2+c^2=p_2$ $a^3+b^3+c^3=p_3$

Answer has form

$\displaystyle S=\sqrt{\frac{Ap_1^2p_2+Bp_1p_3+Cp_1^4}{D}}$

Give answer as $A+B+C+D$ , here $A,B,C,D$ - positive and negative integers.

Task from Boltyansky V.G., Vilenkin N.Ya.