Integral of (x+1)(x+3) dx

1. Rewrite the integrand:

   $$\left(x + 1\right) \left(x + 3\right) = x^{2} + 4 x + 3$$

2. Integrate term-by-term:

   1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

      $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int 4 x\, dx = 4 \int x\, dx$$

      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int x\, dx = \frac{x^{2}}{2}$$

      So, the result is: $2 x^{2}$

   1. The integral of a constant is the constant times the variable of integration:

      $$\int 3\, dx = 3 x$$

   The result is: $\frac{x^{3}}{3} + 2 x^{2} + 3 x$

3. Now simplify:

   $$\frac{x \left(x^{2} + 6 x + 9\right)}{3}$$

4. Add the constant of integration:

   $$\frac{x \left(x^{2} + 6 x + 9\right)}{3}+ \mathrm{constant}$$

The answer is:

$$\frac{x \left(x^{2} + 6 x + 9\right)}{3}+ \mathrm{constant}$$