Integral of ((5x^2-2x+1)^3)(5x-1)dx dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \left(5 x^{2} - 2 x\right) + 1$.

      Then let $du = \left(10 x - 2\right) dx$ and substitute $\frac{du}{2}$:

      $$\int \frac{u^{3}}{2}\, du$$

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

         $$\int u^{3}\, du = \frac{\int u^{3}\, du}{2}$$

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

            $$\int u^{3}\, du = \frac{u^{4}}{4}$$

         So, the result is: $\frac{u^{4}}{8}$

      Now substitute $u$ back in:

      $$\frac{\left(\left(5 x^{2} - 2 x\right) + 1\right)^{4}}{8}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(5 x - 1\right) \left(\left(5 x^{2} - 2 x\right) + 1\right)^{3} = 625 x^{7} - 875 x^{6} + 825 x^{5} - 475 x^{4} + 203 x^{3} - 57 x^{2} + 11 x - 1$$

   2. Integrate term-by-term:

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

         $$\int 625 x^{7}\, dx = 625 \int x^{7}\, dx$$

         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: $\frac{625 x^{8}}{8}$

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

         $$\int \left(- 875 x^{6}\right)\, dx = - 875 \int x^{6}\, dx$$

         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: $- 125 x^{7}$

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

         $$\int 825 x^{5}\, dx = 825 \int x^{5}\, dx$$

         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: $\frac{275 x^{6}}{2}$

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

         $$\int \left(- 475 x^{4}\right)\, dx = - 475 \int x^{4}\, dx$$

         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: $- 95 x^{5}$

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

         $$\int 203 x^{3}\, dx = 203 \int x^{3}\, dx$$

         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: $\frac{203 x^{4}}{4}$

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

         $$\int \left(- 57 x^{2}\right)\, dx = - 57 \int x^{2}\, dx$$

         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: $- 19 x^{3}$

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

         $$\int 11 x\, dx = 11 \int x\, dx$$

         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: $\frac{11 x^{2}}{2}$

      1. The integral of a constant is the constant times the variable of integration:

         $$\int \left(-1\right)\, dx = - x$$

      The result is: $\frac{625 x^{8}}{8} - 125 x^{7} + \frac{275 x^{6}}{2} - 95 x^{5} + \frac{203 x^{4}}{4} - 19 x^{3} + \frac{11 x^{2}}{2} - x$

2. Now simplify:

   $$\frac{\left(5 x^{2} - 2 x + 1\right)^{4}}{8}$$

3. Add the constant of integration:

   $$\frac{\left(5 x^{2} - 2 x + 1\right)^{4}}{8}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(5 x^{2} - 2 x + 1\right)^{4}}{8}+ \mathrm{constant}$$