Evaluate: `int(a x+b)^3dx`

Answer

To evaluate the integral \(\int (ax + b)^3 \, dx\), we can follow these steps:

### Step 1: Identify the integral
We start with the integral:
\[ I = \int (ax + b)^3 \, dx \]

### Step 2: Use the power rule for integration
We can apply the power rule of integration, which states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
In our case, we will treat \(ax + b\) as a single variable, say \(u\), where \(u = ax + b\). Thus, we can rewrite the integral as:
\[ I = \int u^3 \, dx \]

### Step 3: Change of variables
To apply the power rule, we need to express \(dx\) in terms of \(du\). We differentiate \(u\):
\[ \frac{du}{dx} = a \implies dx = \frac{du}{a} \]
Now substituting \(u\) and \(dx\) into the integral gives:
\[ I = \int u^3 \cdot \frac{du}{a} \]

### Step 4: Integrate
Now we can integrate:
\[ I = \frac{1}{a} \int u^3 \, du = \frac{1}{a} \cdot \left( \frac{u^{4}}{4} \right) + C \]
Substituting back \(u = ax + b\):
\[ I = \frac{1}{a} \cdot \frac{(ax + b)^{4}}{4} + C \]

### Step 5: Simplify the expression
This simplifies to:
\[ I = \frac{(ax + b)^{4}}{4a} + C \]

### Final Answer
Thus, the evaluated integral is:
\[ \int (ax + b)^3 \, dx = \frac{(ax + b)^{4}}{4a} + C \]

---