Integral of (8x-12)(4x^2-12x)^4 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 4 x^{2} - 12 x$.

      Then let $du = \left(8 x - 12\right) dx$ and substitute $du$:

      $$\int u^{4}\, du$$

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

         $$\int u^{4}\, du = \frac{u^{5}}{5}$$

      Now substitute $u$ back in:

      $$\frac{\left(4 x^{2} - 12 x\right)^{5}}{5}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(8 x - 12\right) \left(4 x^{2} - 12 x\right)^{4} = 2048 x^{9} - 27648 x^{8} + 147456 x^{7} - 387072 x^{6} + 497664 x^{5} - 248832 x^{4}$$

   2. Integrate term-by-term:

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

         $$\int 2048 x^{9}\, dx = 2048 \int x^{9}\, dx$$

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

            $$\int x^{9}\, dx = \frac{x^{10}}{10}$$

         So, the result is: $\frac{1024 x^{10}}{5}$

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

         $$\int \left(- 27648 x^{8}\right)\, dx = - 27648 \int x^{8}\, dx$$

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

            $$\int x^{8}\, dx = \frac{x^{9}}{9}$$

         So, the result is: $- 3072 x^{9}$

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

         $$\int 147456 x^{7}\, dx = 147456 \int x^{7}\, dx$$

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

            $$\int x^{7}\, dx = \frac{x^{8}}{8}$$

         So, the result is: $18432 x^{8}$

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

         $$\int \left(- 387072 x^{6}\right)\, dx = - 387072 \int x^{6}\, dx$$

         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: $- 55296 x^{7}$

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

         $$\int 497664 x^{5}\, dx = 497664 \int x^{5}\, dx$$

         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: $82944 x^{6}$

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

         $$\int \left(- 248832 x^{4}\right)\, dx = - 248832 \int x^{4}\, dx$$

         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: $- \frac{248832 x^{5}}{5}$

      The result is: $\frac{1024 x^{10}}{5} - 3072 x^{9} + 18432 x^{8} - 55296 x^{7} + 82944 x^{6} - \frac{248832 x^{5}}{5}$

2. Now simplify:

   $$\frac{1024 x^{5} \left(x - 3\right)^{5}}{5}$$

3. Add the constant of integration:

   $$\frac{1024 x^{5} \left(x - 3\right)^{5}}{5}+ \mathrm{constant}$$

The answer is:

$$\frac{1024 x^{5} \left(x - 3\right)^{5}}{5}+ \mathrm{constant}$$