The integral provided is:

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

To solve this, we'll attempt to simplify the integrand using substitution and known integral techniques. One approach could involve substitution due to the presence of $e^x + x + 1$ in the denominator. Here's a possible path:

Step-by-step solution:

1. Substitute: Let $u = e^x + x + 1$.

   • Then, $du = (e^x + 1) \, dx$.
   • This substitution should simplify the denominator, but the expression $x^2 + x$ in the numerator will require rewriting in terms of $u$ to proceed further.

2. Handling the numerator: We need to express $x$ in terms of $u$, and attempt integration after substitution.

However, this process is intricate and might involve advanced techniques or transformations that are not immediately apparent for manual integration.

It appears that this integral does not have a simple elementary solution, but it can potentially be solved using numerical integration methods or more advanced symbolic computation software.

Would you like a detailed breakdown of the substitution steps, or would you prefer a numerical approach for approximating the integral?

Related questions:

1. Would you like to explore a substitution method in more detail?
2. Are you familiar with numerical integration approaches, and would you like help with that?
3. How familiar are you with non-elementary integrals, and should we explore when it's best to switch to numerical methods?
4. Do you want to see similar integrals solved with substitution methods?
5. Would you like to verify if this integral has a known closed-form solution?

Tip:

When tackling complex integrals, recognizing patterns (like derivatives of composite functions) can sometimes simplify the problem significantly.