Integral of x(3x+2)^5 dx

1. Rewrite the integrand:

   $$x \left(3 x + 2\right)^{5} = 243 x^{6} + 810 x^{5} + 1080 x^{4} + 720 x^{3} + 240 x^{2} + 32 x$$

2. Integrate term-by-term:

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

      $$\int 243 x^{6}\, dx = 243 \int x^{6}\, dx$$

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

         $$\int x^{6}\, dx = \frac{x^{7}}{7}$$

      So, the result is: $\frac{243 x^{7}}{7}$

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

      $$\int 810 x^{5}\, dx = 810 \int x^{5}\, dx$$

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

         $$\int x^{5}\, dx = \frac{x^{6}}{6}$$

      So, the result is: $135 x^{6}$

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

      $$\int 1080 x^{4}\, dx = 1080 \int x^{4}\, dx$$

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

         $$\int x^{4}\, dx = \frac{x^{5}}{5}$$

      So, the result is: $216 x^{5}$

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

      $$\int 720 x^{3}\, dx = 720 \int x^{3}\, dx$$

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

         $$\int x^{3}\, dx = \frac{x^{4}}{4}$$

      So, the result is: $180 x^{4}$

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

      $$\int 240 x^{2}\, dx = 240 \int x^{2}\, dx$$

      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: $80 x^{3}$

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

      $$\int 32 x\, dx = 32 \int x\, dx$$

      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: $16 x^{2}$

   The result is: $\frac{243 x^{7}}{7} + 135 x^{6} + 216 x^{5} + 180 x^{4} + 80 x^{3} + 16 x^{2}$

3. Now simplify:

   $$\frac{x^{2} \left(243 x^{5} + 945 x^{4} + 1512 x^{3} + 1260 x^{2} + 560 x + 112\right)}{7}$$

4. Add the constant of integration:

   $$\frac{x^{2} \left(243 x^{5} + 945 x^{4} + 1512 x^{3} + 1260 x^{2} + 560 x + 112\right)}{7}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \left(243 x^{5} + 945 x^{4} + 1512 x^{3} + 1260 x^{2} + 560 x + 112\right)}{7}+ \mathrm{constant}$$