This MathWorld article on the brachistochrone makes the following step in lines (29) and (30):
$$\left( 1 + y \prime ^ 2\right) \left( 1 + \mu y \prime \right) + 2 \left( y - \mu x \right) y \prime \prime = 0$$ can be reduced to: $$\frac{1 + y \prime ^ 2}{\left( 1 + \mu y \prime \right)^2} = \frac{C}{y - \mu x}$$
I found the original journal article from 1995 that the article was based off of, and it doesn't show any more work but it says that this can be solved "through two substitutions and a partial fractions integration". I'm not sure how to approach this; I'm asking for any ideas and preferably a detailed solution if possible. Thanks!