## Step 1: Define the symbolic representation of the problem
The problem involves maximizing an objective function subject to several constraints. Let's define the symbolic variables:
- $x_1$ represents 'milligrams of vitamin B5'
- $x_2$ represents 'milligrams of potassium'
- $x_3$ represents 'milligrams of vitamin E'

## Step 2: Translate the objective function into symbolic notation
The objective function to maximize is $4.9x_1^2 + 9.78x_1x_2 + 5.23x_1x_3 + 4.04x_3^2$.

## 3: List the constraints in symbolic notation
1. $x_1$ has a muscle growth index of 2.14: $2.14x_1$
2. The kidney support index of $x_1$ is 1.45: $1.45x_1$
3. The muscle growth index of $x_2$ is 0.0: $0.0x_2$
4. $x_2$ has a kidney support index of 4.45: $4.45x_2$
5. $x_3$ has a muscle growth index of 4.34: $4.34x_3$
6. $x_3$ has a kidney support index of 3.57: $3.57x_3$
7. $x_2^2 + x_3^2 \geq 11$
8. $2.14x_1 + 4.34x_3 \geq 19$
9. $(2.14x_1)^2 + (0.0x_2)^2 + (4.34x_3)^2 \geq 16$
10. $(2.14x_1)^2 + (0.0x_2)^2 \leq 33$
11. $2.14x_1 + 4.34x_3 \leq 57$
12. $2.14x_1 + 0.0x_2 + 4.34x_3 \leq 57$
13. $(1.45x_1)^2 + (3.57x_3)^2 \leq 41$
14. $1.45x_1 + 4.45x_2 \leq 20$
15. $(1.45x_1)^2 + (4.45x_2)^2 + (3.57x_3)^2 \leq 24$
16. $1.45x_1 + 4.45x_2 + 3.57x_3 \leq 24$

## 4: Specify variable types
- $x_1$ can be a non-integer
- $x_2$ must be an integer
- $x_3$ must be an integer

## 5: Write down the symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'milligrams of vitamin B5'), ('x2', 'milligrams of potassium'), ('x3', 'milligrams of vitamin E')],
'objective_function': '4.9*x1^2 + 9.78*x1*x2 + 5.23*x1*x3 + 4.04*x3^2',
'constraints': [
    'x2^2 + x3^2 >= 11',
    '2.14*x1 + 4.34*x3 >= 19',
    '(2.14*x1)^2 + (4.34*x3)^2 >= 16',
    '(2.14*x1)^2 <= 33',
    '2.14*x1 + 4.34*x3 <= 57',
    '2.14*x1 + 4.34*x3 <= 57',
    '(1.45*x1)^2 + (3.57*x3)^2 <= 41',
    '1.45*x1 + 4.45*x2 <= 20',
    '(1.45*x1)^2 + (4.45*x2)^2 + (3.57*x3)^2 <= 24',
    '1.45*x1 + 4.45*x2 + 3.57*x3 <= 24'
]
}
```

## 6: Implement the problem in Gurobi
```python
import gurobi

def optimize_problem():
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x1")  # milligrams of vitamin B5
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x2", vtype=gurobi.GRB.INTEGER)  # milligrams of potassium
    x3 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x3", vtype=gurobi.GRB.INTEGER)  # milligrams of vitamin E

    # Objective function
    model.setObjective(4.9*x1**2 + 9.78*x1*x2 + 5.23*x1*x3 + 4.04*x3**2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x2**2 + x3**2 >= 11)
    model.addConstr(2.14*x1 + 4.34*x3 >= 19)
    model.addConstr((2.14*x1)**2 + (4.34*x3)**2 >= 16)
    model.addConstr((2.14*x1)**2 <= 33)
    model.addConstr(2.14*x1 + 4.34*x3 <= 57)
    model.addConstr(2.14*x1 + 4.34*x3 <= 57)
    model.addConstr((1.45*x1)**2 + (3.57*x3)**2 <= 41)
    model.addConstr(1.45*x1 + 4.45*x2 <= 20)
    model.addConstr((1.45*x1)**2 + (4.45*x2)**2 + (3.57*x3)**2 <= 24)
    model.addConstr(1.45*x1 + 4.45*x2 + 3.57*x3 <= 24)

    # Solve the problem
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"milligrams of vitamin B5: {x1.varValue}")
        print(f"milligrams of potassium: {x2.varValue}")
        print(f"milligrams of vitamin E: {x3.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```