`I = Int ( x^(2) + 5)^(3) dx`.

Answer

To solve the integral \( I = \int (x^2 + 5)^3 \, dx \), we will use the binomial expansion and then integrate term by term. Here is a step-by-step solution:

### Step 1: Expand the integrand using the binomial theorem
We know that \( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \). In our case, let \( a = x^2 \) and \( b = 5 \).

\[ (x^2 + 5)^3 = (x^2)^3 + 3(x^2)^2(5) + 3(x^2)(5^2) + (5)^3 \]

Calculating each term:
- \( (x^2)^3 = x^6 \)
- \( 3(x^2)^2(5) = 15x^4 \)
- \( 3(x^2)(25) = 75x^2 \)
- \( (5)^3 = 125 \)

Thus, we have:
\[ (x^2 + 5)^3 = x^6 + 15x^4 + 75x^2 + 125 \]

### Step 2: Write the integral
Now we can rewrite the integral:
\[ I = \int (x^6 + 15x^4 + 75x^2 + 125) \, dx \]

### Step 3: Integrate term by term
We will integrate each term separately:
1. \( \int x^6 \, dx = \frac{x^{7}}{7} \)
2. \( \int 15x^4 \, dx = 15 \cdot \frac{x^{5}}{5} = 3x^{5} \)
3. \( \int 75x^2 \, dx = 75 \cdot \frac{x^{3}}{3} = 25x^{3} \)
4. \( \int 125 \, dx = 125x \)

Combining these results, we get:
\[ I = \frac{x^{7}}{7} + 3x^{5} + 25x^{3} + 125x + C \]

### Final Answer
Thus, the final result of the integral is:
\[ I = \frac{x^{7}}{7} + 3x^{5} + 25x^{3} + 125x + C \]