To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves identifying the variables, the objective function, and the constraints.

### Symbolic Representation:

- **Variables:**
  - `x1`: milligrams of vitamin B4
  - `x2`: milligrams of calcium
  - `x3`: grams of carbohydrates

- **Objective Function:**
  The objective is to minimize the following expression:
  \(9.36x_1^2 + 9.52x_1x_2 + 9.56x_1x_3 + 7.32x_2^2 + 4.46x_2x_3 + 1.98x_3^2 + 1.78x_1 + 7.27x_2 + 4.88x_3\)

- **Constraints:**
  1. \(6x_1 + 3x_2 \geq 12\)
  2. \(6x_1 + 9x_3 \geq 11\)
  3. \(6x_1 + 3x_2 + 9x_3 \geq 11\)
  4. \(-6x_1^2 + 8x_2^2 \geq 0\)
  5. \(6x_1 + 3x_2 + 9x_3 \leq 49\)
  6. \(x_1\) must be an integer.
  7. \(x_2\) must be an integer.
  8. \(x_3\) can be any real number.

### Symbolic Representation in JSON Format:
```json
{
  'sym_variables': [('x1', 'milligrams of vitamin B4'), ('x2', 'milligrams of calcium'), ('x3', 'grams of carbohydrates')],
  'objective_function': '9.36*x1**2 + 9.52*x1*x2 + 9.56*x1*x3 + 7.32*x2**2 + 4.46*x2*x3 + 1.98*x3**2 + 1.78*x1 + 7.27*x2 + 4.88*x3',
  'constraints': [
    '6*x1 + 3*x2 >= 12',
    '6*x1 + 9*x3 >= 11',
    '6*x1 + 3*x2 + 9*x3 >= 11',
    '-6*x1**2 + 8*x2**2 >= 0',
    '6*x1 + 3*x2 + 9*x3 <= 49'
  ]
}
```

### Gurobi Code:
```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="milligrams_of_vitamin_B4")
x2 = m.addVar(vtype=GRB.INTEGER, name="milligrams_of_calcium")
x3 = m.addVar(vtype=GRB.CONTINUOUS, name="grams_of_carbohydrates")

# Define the objective function
m.setObjective(9.36*x1**2 + 9.52*x1*x2 + 9.56*x1*x3 + 7.32*x2**2 + 4.46*x2*x3 + 1.98*x3**2 + 1.78*x1 + 7.27*x2 + 4.88*x3, GRB.MINIMIZE)

# Add constraints
m.addConstr(6*x1 + 3*x2 >= 12, "immune_support_index_1")
m.addConstr(6*x1 + 9*x3 >= 11, "immune_support_index_2")
m.addConstr(6*x1 + 3*x2 + 9*x3 >= 11, "total-immune-support-index")
m.addConstr(-6*x1**2 + 8*x2**2 >= 0, "quadratic_constraint")
m.addConstr(6*x1 + 3*x2 + 9*x3 <= 49, "upper_bound_total_immunity")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of vitamin B4: {x1.x}")
    print(f"Milligrams of calcium: {x2.x}")
    print(f"Grams of carbohydrates: {x3.x}")
else:
    print("No optimal solution found")
```