Let's start with Pressure Profile in Homogeneous Steady-State Pipe Flow @model:

\left[\rho(p) -  j_m^2 \cdot c(p)   \right]  \frac{dp}{dl} = \rho^2(p) \, g \, \cos \theta(l)  - \frac{ j_m^2 }{2 d} \cdot  f(p)

p(l=0) = p_0

and assume constant pipe inclination:

\theta(l) = \theta = \rm const

Let's define constant number:

G = g \cdot \cos \theta = \rm const

and rewrite the equation as:

\frac{\left[\rho(p) -  j_m^2 \cdot c(p)   \right] \, dp}{\rho^2(p) \, G  - \frac{ j_m^2 }{2 d} \cdot  f(p)} = dl

The integration of the left side of with the boundary condition leads to:

L =\int_{p_0}^{p} \frac{ \rho(p) - j_m^2 \, c(p) }{G \, \rho^2(p) - F(\rho(p))} \, dp

where

This can be further re-written as:

L = \int_{p_0}^{p} \frac{ \rho \, dp}{G \, \rho^2 - F(\rho)} 
- j_m^2 \cdot \int_{\rho_0}^{\rho} \frac{1}{\rho} \, \frac{d \rho}{G \, \rho^2 - F(\rho)}

or

L =\int_{\rho_0}^{\rho} \frac{ 1/c(\rho) - j_m^2/\rho  }{G \, \rho^2 - F(\rho)} \, d\rho

See also