In[1]:= a = 2;

In[2]:= b[t_] = 3/10 + I + 5/2 Exp[I t];

In[3]:= z[t_] = 1/2 (b[t] + a^2/b[t])

Out[3]= 1/2 ((3/10 + I) + (5 E^(I t))/2 + 
   4/((3/10 + I) + (5 E^(I t))/2))

In[7]:= reimz[t_] = {Re[z[t]], Im[z[t]]} // ComplexExpand // 
  FullSimplify

Out[7]= {1/
  20 (3 + 25 Cos[t]) (1 + 200/(367 + 75 Cos[t] + 250 Sin[t])), -(1/
   2) + (5 Sin[t])/4 + (267 + 75 Cos[t])/(
  367 + 75 Cos[t] + 250 Sin[t])}

In[12]:= dz[t_] = D[reimz[t], t] // FullSimplify

Out[12]= {-((
   250 (3 + 25 Cos[t]) (10 Cos[t] - 3 Sin[t]))/(367 + 75 Cos[t] + 
     250 Sin[t])^2) - 
  5/4 Sin[t] (1 + 200/(367 + 75 Cos[t] + 250 Sin[t])), 
 5/4 (Cos[t] - (
    600 (25 + 89 Cos[t] + 10 Sin[t]))/(367 + 75 Cos[t] + 
      250 Sin[t])^2)}

In[16]:= NIntegrate[Sqrt[dz[t].dz[t]], {t, 0, 2 \[Pi]}]

Out[16]= 9.40235

You can get the area of the curve using differential forms.

In[32]:= tw[t] = TensorWedge[reimz[t], dz[t]][[1, 2]]

Out[32]= ((3 + 25 Cos[t]) (567 + 75 Cos[t] + 250 Sin[t]) (-15000 + 
      81289 Cos[t] + 55050 Cos[t]^2 + 5625 Cos[t]^3 - 6000 Sin[t] + 
      183500 Cos[t] Sin[t] + 37500 Cos[t]^2 Sin[t] + 
      62500 Cos[t] Sin[t]^2))/(16 (367 + 75 Cos[t] + 
      250 Sin[t])^3) + (5 (334 + 150 Cos[t] + 1335 Sin[t] + 
      375 Cos[t] Sin[t] + 1250 Sin[t]^2) (6000 Cos[t] + 
      50000 Cos[t]^2 + 206289 Sin[t] + 55050 Cos[t] Sin[t] + 
      5625 Cos[t]^2 Sin[t] + 233500 Sin[t]^2 + 
      37500 Cos[t] Sin[t]^2 + 62500 Sin[t]^3))/(16 (367 + 75 Cos[t] + 
      250 Sin[t])^3)

In[33]:= 1/2 NIntegrate[tw[t], {t, 0, 2 \[Pi]}]

Out[33]= 1.95895