Equation of a parabola

What is the equation of a parabola that:

1.) Goes through the point (0,0)
2.) Goes through the point (A,B)
3.) Has an arc length L from x = 0 to x = A
4.) Is concave up

It is a function of the form y=ax^2+bx,

where b=(B/A)-aA

and where a is the solution to the equation

\!\(\(1\/\(4\ a\
          A\)\) \((\((a\ A\^2 +
            B)\)\ \@\(\(A\^2\ \((1 + \((aA)\)\^2)\) + 2\
              a\ \(A\^2\) B + B\^2\)\/A\^2\) + \((a\ A\^2 - B)\)\ \@\(1 + \((\
\(-a\)\ A\^2 + B)\)\^2\/A\^2\) + A\ ArcSinh[\(2  a\ A\^2 + B\)\/A] + A\ \
ArcSinh[aA - B\/A])\) == L\)

(just put this into mathematica to make sense of this)