Integral of (2x+1)e^7x dx

1. Rewrite the integrand:

   $$x e^{7} \left(2 x + 1\right) = 2 x^{2} e^{7} + x e^{7}$$

2. Integrate term-by-term:

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int 2 x^{2} e^{7}\, dx = 2 e^{7} \int x^{2}\, dx$$

      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

      So, the result is: $\frac{2 x^{3} e^{7}}{3}$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int x e^{7}\, dx = e^{7} \int x\, dx$$

      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int x\, dx = \frac{x^{2}}{2}$$

      So, the result is: $\frac{x^{2} e^{7}}{2}$

   The result is: $\frac{2 x^{3} e^{7}}{3} + \frac{x^{2} e^{7}}{2}$

3. Now simplify:

   $$\frac{x^{2} \left(4 x + 3\right) e^{7}}{6}$$

4. Add the constant of integration:

   $$\frac{x^{2} \left(4 x + 3\right) e^{7}}{6}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \left(4 x + 3\right) e^{7}}{6}+ \mathrm{constant}$$