I need to find the value of z for a particular value of Dc (eg. 500), but z is inside an integral, and I'm not able to use Solve since the integral is giving Hypergeometric2F1 function as the output.

OmegaM = 0.3111;
OmegaLambda = 0.6889;
Dc = 500;

eqn = Integrate[(OmegaM (1 + z1)^3 + OmegaLambda)^(-1/2), {z1, 0, z}, Assumptions -> z > 0]

(*Output: -1.1473 + (1.20482 + 1.20482 z) Hypergeometric2F1[0.333333, 0.5, 1.33333, -0.451589 (1. + z)^3]*)

zvalue = Solve[eqn == Dc, z]
 
(*Output: Solve was unable to solve the system with inexact coefficients or the 
system obtained by direct rationalization of inexact numbers present
in the system. Since many of the methods used by Solve require exact
input, providing Solve with an exact version of the system may help.*)

Is there any other way I can solve this equation?

Also, Integrate is taking some time and I'd like it to be fast since I need to put it in a loop with lots of z values to be computed for corresponding Dc values.

I need to find the value of $z$ for a particular value of $D_c$ (eg. $500$), but $z$ is inside an integral, and I'm not able to use Solve since the integral is giving Hypergeometric2F1 function as the output.

OmegaM = 0.3111;
OmegaLambda = 0.6889;
Dc = 500;

eqn = Integrate[(OmegaM (1 + z1)^3 + OmegaLambda)^(-1/2), {z1, 0, z}, 
                  Assumptions -> z > 0]

-1.1473+(1.20482+1.20482z)Hypergeometric2F1[0.333333,0.5,1.33333,-0.451589(1.+z)^3]

zvalue = Solve[eqn == Dc, z]

Solve was unable to solve the system with inexact coefficients or the 

system obtained by direct rationalization of inexact numbers present
in the system. Since many of the methods used by Solve require exact
input, providing Solve with an exact version of the system may help.

Is there any other way I can solve this equation?

Also, Integrate is taking some time and I'd like it to be fast since I need to put it in a loop with lots of $z$ values to be computed for corresponding $D_c$ values.