## Step 1: Define the symbolic representation of the variables
The variables are defined as follows:
- $x_1$ represents 'milligrams of vitamin B3'
- $x_2$ represents 'milligrams of vitamin B6'
- $x_3$ represents 'milligrams of vitamin B9'
- $x_4$ represents 'milligrams of vitamin B5'

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is:
\[3x_1^2 + 2x_1x_3 + 7x_1x_4 + x_2^2 + 7x_2x_4 + x_3^2 + 4x_3x_4 + 3x_4^2 + 3x_1 + 3x_2 + 7x_3 + 6x_4\]

## 3: Define the constraints in symbolic notation
The constraints are:
1. $9x_1 \leq 118$
2. $4x_2 \leq 118$
3. $2x_3 \leq 118$
4. $19x_4 \leq 118$
5. $4x_2 + 19x_4 \geq 19$
6. $16x_2^2 + 4x_3^2 \geq 13$
7. $9x_1 + 4x_2 \geq 11$
8. $9x_1 + 19x_4 \geq 18$
9. $9x_1 + 4x_2 + 19x_4 \geq 25$
10. $9x_1 + 2x_3 + 19x_4 \geq 25$
11. $9x_1 + 4x_2 + 19x_4 \geq 21$
12. $4x_2 + 2x_3 + 19x_4 \geq 23$
13. $9x_1 + 2x_3 + 19x_4 \geq 23$
14. $9x_1 + 4x_2 + 19x_4 \geq 23$
15. $9x_1 + 4x_2 + 2x_3 + 19x_4 \geq 23$
16. $81x_1^2 - 7x_4^2 \geq 0$
17. $8x_1 - 8x_2 \geq 0$
18. $9x_1 + 2x_3 \leq 59$

## 4: Consider the variable restrictions
- $x_3$ must be an integer
- $x_4$ must be an integer
- $x_1$ and $x_2$ can be non-integers

## 5: Write the Gurobi code
```python
import gurobi

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

    # Define variables
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x1")  # milligrams of vitamin B3
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x2")  # milligrams of vitamin B6
    x3 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x3", vtype=gurobi.GRB.INTEGER)  # milligrams of vitamin B9
    x4 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x4", vtype=gurobi.GRB.INTEGER)  # milligrams of vitamin B5

    # Objective function
    model.setObjective(3*x1**2 + 2*x1*x3 + 7*x1*x4 + x2**2 + 7*x2*x4 + x3**2 + 4*x3*x4 + 3*x4**2 + 3*x1 + 3*x2 + 7*x3 + 6*x4, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(9*x1 <= 118)
    model.addConstr(4*x2 <= 118)
    model.addConstr(2*x3 <= 118)
    model.addConstr(19*x4 <= 118)
    model.addConstr(4*x2 + 19*x4 >= 19)
    model.addConstr(16*x2**2 + 4*x3**2 >= 13)
    model.addConstr(9*x1 + 4*x2 >= 11)
    model.addConstr(9*x1 + 19*x4 >= 18)
    model.addConstr(9*x1 + 4*x2 + 19*x4 >= 25)
    model.addConstr(9*x1 + 2*x3 + 19*x4 >= 25)
    model.addConstr(9*x1 + 4*x2 + 19*x4 >= 21)
    model.addConstr(4*x2 + 2*x3 + 19*x4 >= 23)
    model.addConstr(9*x1 + 2*x3 + 19*x4 >= 23)
    model.addConstr(9*x1 + 4*x2 + 19*x4 >= 23)
    model.addConstr(9*x1 + 4*x2 + 2*x3 + 19*x4 >= 23)
    model.addConstr(81*x1**2 - 7*x4**2 >= 0)
    model.addConstr(8*x1 - 8*x2 >= 0)
    model.addConstr(9*x1 + 2*x3 <= 59)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Objective: ", model.objVal)
        print("x1: ", x1.varValue)
        print("x2: ", x2.varValue)
        print("x3: ", x3.varValue)
        print("x4: ", x4.varValue)
    else:
        print("No solution found")

solve_optimization_problem()
```

## 6: Symbolic representation in JSON format
```json
{
    "sym_variables": [
        ["x1", "milligrams of vitamin B3"],
        ["x2", "milligrams of vitamin B6"],
        ["x3", "milligrams of vitamin B9"],
        ["x4", "milligrams of vitamin B5"]
    ],
    "objective_function": "3*x1^2 + 2*x1*x3 + 7*x1*x4 + x2^2 + 7*x2*x4 + x3^2 + 4*x3*x4 + 3*x4^2 + 3*x1 + 3*x2 + 7*x3 + 6*x4",
    "constraints": [
        "9*x1 <= 118",
        "4*x2 <= 118",
        "2*x3 <= 118",
        "19*x4 <= 118",
        "4*x2 + 19*x4 >= 19",
        "16*x2^2 + 4*x3^2 >= 13",
        "9*x1 + 4*x2 >= 11",
        "9*x1 + 19*x4 >= 18",
        "9*x1 + 4*x2 + 19*x4 >= 25",
        "9*x1 + 2*x3 + 19*x4 >= 25",
        "9*x1 + 4*x2 + 19*x4 >= 21",
        "4*x2 + 2*x3 + 19*x4 >= 23",
        "9*x1 + 2*x3 + 19*x4 >= 23",
        "9*x1 + 4*x2 + 19*x4 >= 23",
        "9*x1 + 4*x2 + 2*x3 + 19*x4 >= 23",
        "81*x1^2 - 7*x4^2 >= 0",
        "8*x1 - 8*x2 >= 0",
        "9*x1 + 2*x3 <= 59"
    ]
}
```