Observe the integral:\[I = \int \frac{x + 1}{x(1 + xe^x)} \, dx.\]Step 1: Employ substitution. Let \( t = xe^x \). Consequently, \( x + 1 = t \), and \( dx = \frac{dt}{e^x} \).Step 2: Reformulate the integral. Upon substitution, the integral becomes:\[I = \int \frac{dt}{t(1 + t)}.\]Step 3: Decompose the integrand. Separate the integrand into simpler fractions:\[I = \int \left( \frac{1}{t} - \frac{1}{1 + t} \right) \, dt.\]Step 4: Perform integration. Integrate each term:\[I = \log |t| - \log |1 + t| + C.\]Step 5: Revert to the original variable. Substitute \( t = xe^x \) back into the expression:\[I = \log \left| \frac{xe^x}{1 + xe^x} \right| + C.\] Final Answer:\[\boxed{\log \left| \frac{xe^x}{1 + xe^x} \right| + C}\]