Here's the Gurobi code for the optimization problem:

```python
import gurobipy as gp

# Create a new model
m = gp.Model("vitamin_optimization")

# Create variables
vitamin_k = m.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="vitamin_k")
vitamin_b5 = m.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="vitamin_b5")
vitamin_a = m.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="vitamin_a")

# Set objective function
m.setObjective(3 * vitamin_k * vitamin_b5 + 3 * vitamin_b5 * vitamin_b5 + 2 * vitamin_b5 + 3 * vitamin_a, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(5 * vitamin_k + 4 * vitamin_b5 + 14 * vitamin_a <= 132, "r0") # cognitive performance index
m.addConstr(17 * vitamin_k + 10 * vitamin_b5 + 5 * vitamin_a <= 129, "r1") # kidney support index
m.addConstr(14 * vitamin_k + 17 * vitamin_b5 + 4 * vitamin_a <= 116, "r2") # cardiovascular support index
m.addConstr(10 * vitamin_k + 16 * vitamin_b5 + 6 * vitamin_a <= 231, "r3") # immune support index

m.addConstr(4 * vitamin_b5 + 14 * vitamin_a >= 42, "c1")
m.addConstr(5 * vitamin_k + 14 * vitamin_a >= 38, "c2")
m.addConstr(5 * vitamin_k + 4 * vitamin_b5 + 14 * vitamin_a >= 32, "c3")
m.addConstr(17 * vitamin_k * vitamin_k + 5 * vitamin_a * vitamin_a >= 30, "c4")
m.addConstr(14 * vitamin_k * vitamin_k + 17 * vitamin_b5 * vitamin_b5 + 4 * vitamin_a * vitamin_a >= 27, "c5")
m.addConstr(10 * vitamin_k + 16 * vitamin_b5 + 6 * vitamin_a >= 59, "c6")

m.addConstr(4 * vitamin_b5 + 14 * vitamin_a <= 70, "c7")
m.addConstr(5 * vitamin_k + 4 * vitamin_b5 + 14 * vitamin_a <= 70, "c8")
m.addConstr(10 * vitamin_b5 * vitamin_b5 + 5 * vitamin_a * vitamin_a <= 105, "c9")
m.addConstr(17 * vitamin_k + 10 * vitamin_b5 <= 121, "c10")
m.addConstr(17 * vitamin_k + 10 * vitamin_b5 + 5 * vitamin_a <= 121, "c11")
m.addConstr(14 * vitamin_k + 17 * vitamin_b5 <= 68, "c12")
m.addConstr(14 * vitamin_k + 4 * vitamin_a <= 72, "c13")
m.addConstr(14 * vitamin_k + 17 * vitamin_b5 + 4 * vitamin_a <= 72, "c14")
m.addConstr(10 * vitamin_k + 6 * vitamin_a <= 85, "c15")
m.addConstr(10 * vitamin_k + 16 * vitamin_b5 <= 102, "c16")
m.addConstr(10 * vitamin_k + 16 * vitamin_b5 + 6 * vitamin_a <= 102, "c17")


# Optimize model
m.optimize()

# Print results
if m.status == gp.GRB.OPTIMAL:
    print('Obj: %g' % m.objVal)
    print('vitamin_k:', vitamin_k.x)
    print('vitamin_b5:', vitamin_b5.x)
    print('vitamin_a:', vitamin_a.x)
elif m.status == gp.GRB.INFEASIBLE:
    print('The model is infeasible.')
else:
    print('Optimization ended with status %d' % m.status)
```
