Until today I thought that $$ \int_0^b \sqrt{\frac{1+y'(x)^2}{2gy(x)}} dx$$ would be the only functional to derive the brachystochrone, but in the textbook Variational Methods in Mathematical Physics by Blanchard and Brünining I found the following integral: $$ \int_0^b \sqrt{\frac{1+y'(x)^2}{2gx}} dx $$
Is this possible? If yes, how can one derive this equation?
I might have found something that could have to do with this but I am not sure whether it is the right reasoning, namely: $ds^2=\frac{dx^2+dy^2}{2x}$ and $ds^2=\frac{dx^2+dy^2}{2y}$ are both line elements for a cycloidal metric.