To solve the integral:

$\int e^x(1 + x)(x + \ln x) \, dx$

we need to approach it step by step.

Step 1: Expand the expression inside the integral.

We begin by expanding the product $(1 + x)(x + \ln x)$:

$(1 + x)(x + \ln x) = 1 \cdot (x + \ln x) + x \cdot (x + \ln x)$ $= x + \ln x + x^2 + x \ln x$

Now the integral becomes:

$\int e^x \left( x + \ln x + x^2 + x \ln x \right) \, dx$

Step 2: Break the integral into separate parts.

We can now separate the integral into simpler parts:

$\int e^x x \, dx + \int e^x \ln x \, dx + \int e^x x^2 \, dx + \int e^x x \ln x \, dx$

Step 3: Solve the integrals.

Each of these integrals involves the exponential function $e^x$, which suggests that integration by parts will be helpful. Let's solve these one by one.

1. $\int e^x x \, dx$

Using integration by parts, where we set $u = x$ and $dv = e^x \, dx$, we have:

$du = dx, \quad v = e^x$ $\int e^x x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C_1$

2. $\int e^x \ln x \, dx$

This one requires integration by parts, where we set $u = \ln x$ and $dv = e^x \, dx$. Then:

$du = \frac{1}{x} dx, \quad v = e^x$ $\int e^x \ln x \, dx = \ln x \cdot e^x - \int \frac{e^x}{x} \, dx$

This integral doesn't have a simple elementary form (the second integral $\int \frac{e^x}{x} \, dx$ is the Exponential Integral, denoted by $\text{Ei}(x)$).

3. $\int e^x x^2 \, dx$

Using integration by parts again, where we set $u = x^2$ and $dv = e^x \, dx$, we get:

$du = 2x \, dx, \quad v = e^x$ $\int e^x x^2 \, dx = x^2 e^x - \int 2x e^x \, dx$

We already know how to solve $\int e^x x \, dx$, so substitute that result here:

$= x^2 e^x - 2(x e^x - e^x)$ $= e^x (x^2 - 2x + 2) + C_2$

4. $\int e^x x \ln x \, dx$

Again using integration by parts with $u = x \ln x$ and $dv = e^x \, dx$, we have:

$du = \ln x \, dx + \frac{1}{x} dx, \quad v = e^x$ $\int e^x x \ln x \, dx = x \ln x \cdot e^x - \int e^x \left( \ln x + \frac{1}{x} \right) \, dx$

This is another complex integral involving the Exponential Integral function.

Step 4: Combine the results.

Combining all the results, the integral becomes:

$\int e^x(1 + x)(x + \ln x) \, dx = \text{(Result from the expansion above)}$

However, parts of the integral (involving $\ln x$ and $\frac{e^x}{x}$) lead to special functions that are outside elementary functions.

Would you like a deeper breakdown or further clarification on any of the steps above? Or perhaps an estimate for the remaining terms?