Euler's Number!

There is a curve such that the middle point of the normal from any point on the curve to the $x$ -axis lies on the parabola $2y^2=x$ . The equation of the curve is:

$y^2 = bx + c + de^{nx}$

for some constants $b,c,d,n$ , where e is Euler's number .

What is the value of $b + c + d + n$ ?