Equilateral Triangle Inside a Square

An equilateral triangle is inscribed inside a square as shown. If the area of the two smaller triangles are 22 and 23, find the area of the shaded quadrilateral. The answer is in the form $a+b\sqrt{c}$ where $a$, $b$, $c$ are positive integers and $c$ is square-free. Input the answer as the sum $a+b+c$.