## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'milligrams of vitamin A' and 'milligrams of vitamin B5'. Let's denote 'milligrams of vitamin A' as $x_1$ and 'milligrams of vitamin B5' as $x_2$. The objective function to minimize is $1.66x_1^2 + 2.23x_2^2 + 5.28x_2$.

## Step 2: Translate the given constraints into symbolic notation
The constraints given are:
- $20x_1 + 1x_2 \geq 32$ (immune support index constraint)
- $13x_1 + 5x_2 \geq 11$ (cardiovascular support index constraint)
- $5x_1^2 - 4x_2^2 \geq 0$ (quadratic constraint)
- $20x_1 + 1x_2 \leq 54$ (upper bound for immune support index)
- $13x_1 + 5x_2 \leq 43$ (upper bound for cardiovascular support index)
- $x_1$ is a whole number (integer constraint)
- $x_2$ is a real number (no constraint needed in Gurobi as it's the default)

## 3: Express the problem in a symbolic representation
The symbolic representation of the problem is:
- Variables: $x_1$ ('milligrams of vitamin A'), $x_2$ ('milligrams of vitamin B5')
- Objective function: $1.66x_1^2 + 2.23x_2^2 + 5.28x_2$
- Constraints:
  1. $20x_1 + x_2 \geq 32$
  2. $13x_1 + 5x_2 \geq 11$
  3. $5x_1^2 - 4x_2^2 \geq 0$
  4. $20x_1 + x_2 \leq 54$
  5. $13x_1 + 5x_2 \leq 43$
  6. $x_1 \in \mathbb{Z}$ (integer constraint)

## 4: Convert the problem into Gurobi code
```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define variables
x1 = model.addVar(vtype=gurobi.GRB.INTEGER, name='milligrams_of_vitamin_A')
x2 = model.addVar(vtype=gurobi.GRB.CONTINUOUS, name='milligrams_of_vitamin_B5')

# Objective function
model.setObjective(1.66 * x1**2 + 2.23 * x2**2 + 5.28 * x2, gurobi.GRB.MINIMIZE)

# Constraints
model.addConstr(20 * x1 + x2 >= 32, name='immune_support_index_min')
model.addConstr(13 * x1 + 5 * x2 >= 11, name='cardiovascular_support_index_min')
model.addConstr(5 * x1**2 - 4 * x2**2 >= 0, name='quadratic_constraint')
model.addConstr(20 * x1 + x2 <= 54, name='immune_support_index_max')
model.addConstr(13 * x1 + 5 * x2 <= 43, name='cardiovascular_support_index_max')

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print('Optimal solution found.')
    print(f'milligrams of vitamin A: {x1.varValue}')
    print(f'milligrams of vitamin B5: {x2.varValue}')
    print(f'Objective: {model.objVal}')
else:
    print('No optimal solution found.')
```

## 5: Provide the symbolic representation in the required format
```json
{
    'sym_variables': [('x1', 'milligrams of vitamin A'), ('x2', 'milligrams of vitamin B5')],
    'objective_function': '1.66*x1^2 + 2.23*x2^2 + 5.28*x2',
    'constraints': [
        '20*x1 + x2 >= 32',
        '13*x1 + 5*x2 >= 11',
        '5*x1^2 - 4*x2^2 >= 0',
        '20*x1 + x2 <= 54',
        '13*x1 + 5*x2 <= 43',
        'x1 is an integer'
    ]
}
```