To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the given conditions.

Let's denote:
- $x_1$ as the number of slow balls hit,
- $x_2$ as the number of fast balls hit.

The objective is to maximize the total points earned from hitting these balls, where each slow ball contributes 3 points and each fast ball contributes 5 points. Thus, the objective function can be written as:
\[ \text{Maximize: } 3x_1 + 5x_2 \]

The constraints based on the problem description are:
1. Hit at least 5 slow balls: $x_1 \geq 5$
2. Hit at least 3 fast balls: $x_2 \geq 3$
3. Hit at most 8 slow balls: $x_1 \leq 8$
4. Hit at most 8 fast balls: $x_2 \leq 8$
5. Hit no more than 12 balls in total: $x_1 + x_2 \leq 12$

All variables are non-negative since you cannot hit a negative number of balls.

The symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'number of slow balls hit'), ('x2', 'number of fast balls hit')],
    'objective_function': '3*x1 + 5*x2',
    'constraints': ['x1 >= 5', 'x2 >= 3', 'x1 <= 8', 'x2 <= 8', 'x1 + x2 <= 12']
}
```

To solve this problem using Gurobi in Python, we'll use the following code:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="slow_balls")
x2 = m.addVar(vtype=GRB.INTEGER, name="fast_balls")

# Set objective function
m.setObjective(3*x1 + 5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 >= 5, "min_slow_balls")
m.addConstr(x2 >= 3, "min_fast_balls")
m.addConstr(x1 <= 8, "max_slow_balls")
m.addConstr(x2 <= 8, "max_fast_balls")
m.addConstr(x1 + x2 <= 12, "total_max_balls")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hit {x1.x} slow balls and {x2.x} fast balls.")
    print(f"Total points: {3*x1.x + 5*x2.x}")
else:
    print("No optimal solution found.")

```