1. **State the problem:** We need to solve the integral $$\int \frac{\sqrt{x} - x^3 e^x + x^2}{x^3} \, dx.$$\n\n2. **Rewrite the integrand:** Divide each term in the numerator by $x^3$:\n$$\frac{\sqrt{x}}{x^3} - \frac{x^3 e^x}{x^3} + \frac{x^2}{x^3} = x^{\frac{1}{2} - 3} - e^x + x^{2 - 3} = x^{-\frac{5}{2}} - e^x + x^{-1}.$$\n\n3. **Integral becomes:** $$\int \left(x^{-\frac{5}{2}} - e^x + x^{-1}\right) dx = \int x^{-\frac{5}{2}} dx - \int e^x dx + \int x^{-1} dx.$$\n\n4. **Integrate each term:**\n- For $\int x^{-\frac{5}{2}} dx$, add 1 to the exponent: $-\frac{5}{2} + 1 = -\frac{3}{2}$. So,\n$$\int x^{-\frac{5}{2}} dx = \frac{x^{-\frac{3}{2}}}{-\frac{3}{2}} = -\frac{2}{3} x^{-\frac{3}{2}}.$$\n- For $\int e^x dx = e^x.$\n- For $\int x^{-1} dx = \ln|x|.$\n\n5. **Combine results:**\n$$\int \frac{\sqrt{x} - x^3 e^x + x^2}{x^3} dx = -\frac{2}{3} x^{-\frac{3}{2}} - e^x + \ln|x| + C,$$ where $C$ is the constant of integration.