Hi,
I am deriving the universal formulation for time of flight which is in the book 'Fundamentals of Astrodynamics' by Bate, Mueller, & White.
dx=dr/sqrt(-p+2r-(r^2/a)) ----- 1
For e not equal to 1, the indefinite integral calling the constant of integration c_0 is
x+ c_o= sqrt(a)...