To solve the integral of \( x^2 \sqrt{3x + 3} \, dx \), we can use substitution to simplify the expression.

### Step 1: Substitution
Let:
\[
u = 3x + 3
\]
Then:
\[
\frac{du}{dx} = 3 \quad \Rightarrow \quad dx = \frac{du}{3}
\]

We also express \( x \) in terms of \( u \):
\[
x = \frac{u - 3}{3}
\]

### Step 2: Rewrite the integral in terms of \( u \)
Substitute \( x = \frac{u - 3}{3} \) and \( dx = \frac{du}{3} \) into the original integral:
\[
\int x^2 \sqrt{3x + 3} \, dx = \int \left(\frac{u - 3}{3}\right)^2 \sqrt{u} \cdot \frac{du}{3}
\]

### Step 3: Simplify the expression
Simplifying the expression:
\[
= \frac{1}{27} \int (u - 3)^2 \sqrt{u} \, du
\]
Now expand \( (u - 3)^2 \):
\[
(u - 3)^2 = u^2 - 6u + 9
\]
Thus, the integral becomes:
\[
= \frac{1}{27} \int (u^2 - 6u + 9) \sqrt{u} \, du
\]

### Step 4: Break it into simpler integrals
We can now break this into three separate integrals:
\[
= \frac{1}{27} \left( \int u^2 \sqrt{u} \, du - 6 \int u \sqrt{u} \, du + 9 \int \sqrt{u} \, du \right)
\]

### Step 5: Solve each integral
Each of these integrals is a power of \( u \):

1. \( \int u^2 \sqrt{u} \, du = \int u^{\frac{5}{2}} \, du = \frac{2}{7} u^{\frac{7}{2}} \)
2. \( \int u \sqrt{u} \, du = \int u^{\frac{3}{2}} \, du = \frac{2}{5} u^{\frac{5}{2}} \)
3. \( \int \sqrt{u} \, du = \int u^{\frac{1}{2}} \, du = \frac{2}{3} u^{\frac{3}{2}} \)

### Step 6: Substitute back and simplify
Substitute these results back into the expression:
\[
\frac{1}{27} \left( \frac{2}{7} u^{\frac{7}{2}} - 6 \cdot \frac{2}{5} u^{\frac{5}{2}} + 9 \cdot \frac{2}{3} u^{\frac{3}{2}} \right)
\]

Simplifying:
\[
= \frac{2}{189} u^{\frac{7}{2}} - \frac{12}{45} u^{\frac{5}{2}} + \frac{6}{9} u^{\frac{3}{2}}
\]
\[
= \frac{2}{189} u^{\frac{7}{2}} - \frac{4}{15} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}}
\]

### Step 7: Substitute back \( u = 3x + 3 \)
Finally, substitute \( u = 3x + 3 \) back into the solution:
\[
= \frac{2}{189} (3x + 3)^{\frac{7}{2}} - \frac{4}{15} (3x + 3)^{\frac{5}{2}} + \frac{2}{3} (3x + 3)^{\frac{3}{2}} + C
\]

This is the integral of \( x^2 \sqrt{3x + 3} \, dx \).

Would you like more details or have any questions?

### Related Questions:
1. How do you integrate functions with radicals like \( \sqrt{ax + b} \)?
2. What is the substitution method in integration and when to use it?
3. How do you integrate polynomials multiplied by radicals?
4. Can you explain integration by parts for non-polynomial functions?
5. What are other common methods to simplify integrals?

### Tip:
When solving integrals involving roots or powers of linear expressions, substitution is a key technique that can simplify the process.

This guide explains how to solve the integral of x^2 sqrt(3x+3) using substitution. It involves rewriting the integral with u = 3x + 3 and solving through polynomial and radical functions. Suitable for calculus students.

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