Integral of x(5x-1)^5 dx

1. Rewrite the integrand:

   $$x \left(5 x - 1\right)^{5} = 3125 x^{6} - 3125 x^{5} + 1250 x^{4} - 250 x^{3} + 25 x^{2} - x$$

2. Integrate term-by-term:

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

      $$\int 3125 x^{6}\, dx = 3125 \int x^{6}\, dx$$

      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: $\frac{3125 x^{7}}{7}$

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

      $$\int \left(- 3125 x^{5}\right)\, dx = - 3125 \int x^{5}\, dx$$

      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: $- \frac{3125 x^{6}}{6}$

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

      $$\int 1250 x^{4}\, dx = 1250 \int x^{4}\, dx$$

      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: $250 x^{5}$

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

      $$\int \left(- 250 x^{3}\right)\, dx = - 250 \int x^{3}\, dx$$

      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: $- \frac{125 x^{4}}{2}$

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

      $$\int 25 x^{2}\, dx = 25 \int x^{2}\, dx$$

      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: $\frac{25 x^{3}}{3}$

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

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

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

   The result is: $\frac{3125 x^{7}}{7} - \frac{3125 x^{6}}{6} + 250 x^{5} - \frac{125 x^{4}}{2} + \frac{25 x^{3}}{3} - \frac{x^{2}}{2}$

3. Now simplify:

   $$\frac{x^{2} \left(18750 x^{5} - 21875 x^{4} + 10500 x^{3} - 2625 x^{2} + 350 x - 21\right)}{42}$$

4. Add the constant of integration:

   $$\frac{x^{2} \left(18750 x^{5} - 21875 x^{4} + 10500 x^{3} - 2625 x^{2} + 350 x - 21\right)}{42}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \left(18750 x^{5} - 21875 x^{4} + 10500 x^{3} - 2625 x^{2} + 350 x - 21\right)}{42}+ \mathrm{constant}$$