Evaluate
`int(x^(2)+5)^(3)dx`

Answer

To evaluate the integral \( \int (x^2 + 5)^3 \, dx \), we can follow these steps:

### Step 1: Expand the integrand using the binomial theorem
We start by expanding \( (x^2 + 5)^3 \) using the binomial expansion formula:
\[ (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]
Here, let \( a = x^2 \) and \( b = 5 \):
\[ (x^2 + 5)^3 = (x^2)^3 + 5^3 + 3 \cdot x^2 \cdot 5 \cdot (x^2 + 5) \]

### Step 2: Calculate each term
Calculating each term:
- \( (x^2)^3 = x^6 \)
- \( 5^3 = 125 \)
- \( 3 \cdot x^2 \cdot 5 = 15x^2 \)

Substituting these back into the expansion:
\[ (x^2 + 5)^3 = x^6 + 125 + 15x^2(x^2 + 5) \]
Now simplify \( 15x^2(x^2 + 5) \):
\[ 15x^2(x^2 + 5) = 15x^4 + 75x^2 \]

### Step 3: Combine all terms
Now, combine all the terms:
\[ (x^2 + 5)^3 = x^6 + 15x^4 + 75x^2 + 125 \]

### Step 4: Write the integral
Now we can rewrite the integral:
\[ \int (x^2 + 5)^3 \, dx = \int (x^6 + 15x^4 + 75x^2 + 125) \, dx \]

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

### Step 6: Combine the results
Combining all the results, we get:
\[ \int (x^2 + 5)^3 \, dx = \frac{x^7}{7} + 3x^5 + 25x^3 + 125x + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final result is:
\[ \int (x^2 + 5)^3 \, dx = \frac{x^7}{7} + 3x^5 + 25x^3 + 125x + C \]