Solving an Integral equation with uncertainties, by using fsolve and uncertainties packages in Python

Question

I have some variables that are uncertain, these are

w_m = u.ufloat(0.1430, 0.0011)
z_rec = u.ufloat(1089.92, 0.25)
theta_srec = u.ufloat(0.0104110,  0.0000031)
r_srec = u.ufloat(144.43, 0.26)

and some constant values

c = 299792.458  # speed of light in [km/s]
N_eff = 3.046
w_r = 2.469 * 10**(-5) * (1 + (7/8)*(4/11)**(4/3) * N_eff)

My problem is I need to solve an integral defined by this functions

def D_zrec(z):
        return (c/100) / sqrt(w_m * (1+z)**3 + w_r * (1+z)**4 + (h**2 - w_m - w_r))

This function is evaluated for dz but it also contains an unknown h that we need to find, with corresponding uncertainty. So I need to write a code that finds h.

Here is my full code

    from numpy import sqrt, vectorize
from scipy.integrate import quad
import uncertainties as u
from uncertainties.umath import *
from scipy.optimize import fsolve


####  Important Parameters #####
c = 299792.458  # speed of light in [km/s]
N_eff = 3.046
w_r = 2.469 * 10**(-5) * (1 + (7/8)*(4/11)**(4/3) * N_eff)

w_m = u.ufloat(0.1430, 0.0011)
z_rec = u.ufloat(1089.92, 0.25)
theta_srec = u.ufloat(0.0104110,  0.0000031)
r_srec = u.ufloat(144.43, 0.26)

D_zrec_true = r_srec / theta_srec

@u.wrap
def D_zrec_finder(h, w_m, z_rec, D_zrec_true):
    def D_zrec(z):
        return (c/100) / sqrt(w_m * (1+z)**3 + w_r * (1+z)**4 + (h**2 - w_m - w_r))
    result, error = quad(D_zrec, 0, z_rec)
    return D_zrec_true - result

def h0_finder(w_m, z_rec, D_zrec_true):
    vfunc = vectorize(D_zrec_finder)
    sol = fsolve(vfunc, u.ufloat(0.6728, 0.01), args=(w_m, z_rec, D_zrec_true))[0]
    return sol

print(h0_finder(w_m, z_rec, D_zrec_true))

So to summarize I have an integral named as D_zrec that is a function of z, but also contains an unknown number h that we need to find by using fsolve.

I have found 3 sites that might be useful for the coder. Please look at them if you want to help

https://kitchingroup.cheme.cmu.edu/blog/2013/03/07/Another-approach-to-error-propagation/

https://kitchingroup.cheme.cmu.edu/blog/2013/07/10/Uncertainty-in-an-integral-equation/

https://kitchingroup.cheme.cmu.edu/blog/2013/01/23/Solving-integral-equations-with-fsolve/

I have looked at them to write my code but no luck.

Thanks for the help