General Falltime with Air Resistance

I wanted to solve the problem of the time it takes an object to fall when influenced by gravity and quadratic drag, and the best I could do was 4 different formulas that you had to use depending on the initial velocity (greater than 0, between 0 and the terminal velocity, equal to the terminal velocity, or less than the terminal velocity). I wanted to generalize this to a single equation that accounts for all cases (which requires handling complex arguments) and to express it without trig functions by using their definitions involving the natural logarithm, and the final result is an absolute monster. Is there a way to simplify it? The variables v and h refer to their initial values, and k is the constant of proportionality between the object's velocity squared to its acceleration from the drag force. This still is undefined for when the initial velocity is equal to the terminal velocity (-sqrt(g/k)), but the solution to that is fairly trivial (sqrt(k/g) * h).

I can't simplify the difference of squares that you see because it creates problems when you assume only the principle branch, so leaving it expanded was intentional.