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

1. Rewrite the integrand:

   $$\left(x^{2} + 1\right)^{3} = x^{6} + 3 x^{4} + 3 x^{2} + 1$$

2. Integrate term-by-term:

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

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

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

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

      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: $\frac{3 x^{5}}{5}$

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

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

      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: $x^{3}$

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

      $$\int 1\, dx = x$$

   The result is: $\frac{x^{7}}{7} + \frac{3 x^{5}}{5} + x^{3} + x$

3. Add the constant of integration:

   $$\frac{x^{7}}{7} + \frac{3 x^{5}}{5} + x^{3} + x+ \mathrm{constant}$$

The answer is:

$$\frac{x^{7}}{7} + \frac{3 x^{5}}{5} + x^{3} + x+ \mathrm{constant}$$