If you have the symbolic toolbox, then you can use odeToVectorField to convert your ODE into the standard form of H' = f(r, H) needed for ode45. For example
syms h(r)
term1 = (diff(h, r, 2) + 1/r*diff(h, r, 1) + 1/r*diff(h, r, 1)^3)/(1 + diff(h, r, 1)^2)^(3/2);
rhs = 2*r - r^2*h^3*diff(term1, r, 1);
lhs = h/r + diff(h, r, 1);
eq = lhs == rhs;
V = odeToVectorField(eq);
odefun = matlabFunction(V, 'Vars', {'r', 'Y'}); % Y denotes h here.
IC = [0.1; 0.3; 1];
rspan = [0 10];
[r_sol, h_sol] = ode45(odefun, rspan, IC);
However, there seems to be a singularity in the ODE, or there is a typing mistake, the ode45 gives NaN output.
