To find the integral:

$\int \frac{x e^{2x}}{(1 + 2x)^2} \, dx$

we can proceed by using integration by parts and substitution methods. Let's go through the steps:

1. Substitute: Let $u = 1 + 2x$, so $du = 2 \, dx$ or $dx = \frac{du}{2}$. Note that $x = \frac{u - 1}{2}$.

   Under this substitution, the integral becomes:

   = \frac{1}{4} \int \frac{(u - 1) e^{u - 1}}{u^2} \, du$$

2. Simplify the Integral:

   Rewrite the integral as:

   $\frac{1}{4} \int \left( \frac{u - 1}{u^2} \right) e^{u - 1} \, du = \frac{1}{4} \int \left( \frac{1}{u} - \frac{1}{u^2} \right) e^{u - 1} \, du$

3. Integrate by Parts:

   For the term $\int \frac{e^{u - 1}}{u} \, du$, let $v = \frac{1}{u}$ and $dw = e^{u - 1} \, du$. Then proceed with integration by parts.

If you would like, I can continue with these steps in full detail or apply another approach for simpl