## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'airborne infantry companies' and 'engineer platoons', which we can denote as $x_1$ and $x_2$ respectively. The objective function to minimize is $5.75x_1^2 + 5.18x_1x_2 + 3.76x_1$. The constraints are based on the attributes of these variables.

## Step 2: List the constraints in symbolic notation
- Deployment weight: $1.02x_1 + 4.73x_2$
- Defensive capability rating: $0.3x_1 + 1.69x_2$
- Fun factor: $1.34x_1 + 0.4x_2$
- Mobility rating: $2.7x_1 + 0.65x_2$
- Total deployment weight squared: $(1.02x_1)^2 + (4.73x_2)^2 \geq 19$
- Total deployment weight: $1.02x_1 + 4.73x_2 \geq 19$
- Total defensive capability rating: $0.3x_1 + 1.69x_2 \geq 44$
- Total fun factor: $(1.34x_1)^2 + (0.4x_2)^2 \geq 65$ and $1.34x_1 + 0.4x_2 \geq 65$
- Total mobility rating: $2.7x_1 + 0.65x_2 \geq 41$
- Linear constraint: $-8x_1 + 9x_2 \geq 0$
- Combined deployment weight upper bound: $1.02x_1 + 4.73x_2 \leq 78$
- Defensive capability rating squared upper bound: $(0.3x_1)^2 + (1.69x_2)^2 \leq 54$
- Fun factor squared upper bound: $(1.34x_1)^2 + (0.4x_2)^2 \leq 102$
- Mobility rating squared upper bound: $(2.7x_1)^2 + (0.65x_2)^2 \leq 198$
- Integer constraints: $x_1, x_2 \in \mathbb{Z}$

## 3: Convert the problem into a Gurobi model
We will use Gurobi's Python API to model this problem.

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(name="airborne_infantry_companies", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="engineer_platoons", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(5.75 * x1**2 + 5.18 * x1 * x2 + 3.76 * x1, gp.GRB.MINIMIZE)

# Define the constraints
m.addConstr(1.02 * x1 + 4.73 * x2 >= 19, name="total_deployment_weight")
m.addConstr(0.3 * x1 + 1.69 * x2 >= 44, name="total_defensive_capability_rating")
m.addConstr(1.34 * x1 + 0.4 * x2 >= 65, name="total_fun_factor")
m.addConstr(2.7 * x1 + 0.65 * x2 >= 41, name="total_mobility_rating")
m.addConstr(-8 * x1 + 9 * x2 >= 0, name="linear_constraint")
m.addConstr(1.02 * x1 + 4.73 * x2 <= 78, name="combined_deployment_weight_upper_bound")
m.addConstr((0.3 * x1)**2 + (1.69 * x2)**2 <= 54, name="defensive_capability_rating_squared_upper_bound")
m.addConstr((1.34 * x1)**2 + (0.4 * x2)**2 <= 102, name="fun_factor_squared_upper_bound")
m.addConstr((2.7 * x1)**2 + (0.65 * x2)**2 <= 198, name="mobility_rating_squared_upper_bound")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("Airborne Infantry Companies: ", x1.varValue)
    print("Engineer Platoons: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 4: Symbolic Representation
The symbolic representation of the problem is as follows:

```json
{
    'sym_variables': [('x1', 'airborne infantry companies'), ('x2', 'engineer platoons')],
    'objective_function': '5.75*x1^2 + 5.18*x1*x2 + 3.76*x1',
    'constraints': [
        '1.02*x1 + 4.73*x2 >= 19',
        '0.3*x1 + 1.69*x2 >= 44',
        '1.34*x1 + 0.4*x2 >= 65',
        '2.7*x1 + 0.65*x2 >= 41',
        '-8*x1 + 9*x2 >= 0',
        '1.02*x1 + 4.73*x2 <= 78',
        '(0.3*x1)^2 + (1.69*x2)^2 <= 54',
        '(1.34*x1)^2 + (0.4*x2)^2 <= 102',
        '(2.7*x1)^2 + (0.65*x2)^2 <= 198',
        'x1, x2 are integers'
    ]
}
```