SILK-ROAD 2010 Problem 3

For positive real numbers $a, b, c, d,$ satisfying the following conditions: $a(c^2 - 1)=b(b^2+c^2)$ and $d \leq 1$ , prove that : $d(a \sqrt{1-d^2} + b^2 \sqrt{1+d^2}) \leq \frac{(a+b)c}{2}$