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

1. There are multiple ways to do this integral.

   Method #1

   1. Rewrite the integrand:

      $$\frac{3 x + 5}{\left(\left(x^{2} + 2 x\right) + 2\right)^{2}} = \frac{3 x + 5}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}$$

   2. Rewrite the integrand:

      $$\frac{3 x + 5}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4} = \frac{3 x}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4} + \frac{5}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}$$

   3. Integrate term-by-term:

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

         $$\int \frac{3 x}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}\, dx = 3 \int \frac{x}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}\, dx$$

         1. Don't know the steps in finding this integral.

            But the integral is

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

         So, the result is: $- \frac{3 \left(x + 2\right)}{2 x^{2} + 4 x + 4} - \frac{3 \operatorname{atan}{\left(x + 1 \right)}}{2}$

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

         $$\int \frac{5}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}\, dx = 5 \int \frac{1}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}\, dx$$

         1. Don't know the steps in finding this integral.

            But the integral is

            $$\frac{x + 1}{2 x^{2} + 4 x + 4} + \frac{\operatorname{atan}{\left(x + 1 \right)}}{2}$$

         So, the result is: $\frac{5 \left(x + 1\right)}{2 x^{2} + 4 x + 4} + \frac{5 \operatorname{atan}{\left(x + 1 \right)}}{2}$

      The result is: $\frac{5 \left(x + 1\right)}{2 x^{2} + 4 x + 4} - \frac{3 \left(x + 2\right)}{2 x^{2} + 4 x + 4} + \operatorname{atan}{\left(x + 1 \right)}$

   Method #2

   1. Rewrite the integrand:

      $$\frac{3 x + 5}{\left(\left(x^{2} + 2 x\right) + 2\right)^{2}} = \frac{3 x}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4} + \frac{5}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}$$

   2. Integrate term-by-term:

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

         $$\int \frac{3 x}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}\, dx = 3 \int \frac{x}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}\, dx$$

         1. Don't know the steps in finding this integral.

            But the integral is

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

         So, the result is: $- \frac{3 \left(x + 2\right)}{2 x^{2} + 4 x + 4} - \frac{3 \operatorname{atan}{\left(x + 1 \right)}}{2}$

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

         $$\int \frac{5}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}\, dx = 5 \int \frac{1}{x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 4}\, dx$$

         1. Don't know the steps in finding this integral.

            But the integral is

            $$\frac{x + 1}{2 x^{2} + 4 x + 4} + \frac{\operatorname{atan}{\left(x + 1 \right)}}{2}$$

         So, the result is: $\frac{5 \left(x + 1\right)}{2 x^{2} + 4 x + 4} + \frac{5 \operatorname{atan}{\left(x + 1 \right)}}{2}$

      The result is: $\frac{5 \left(x + 1\right)}{2 x^{2} + 4 x + 4} - \frac{3 \left(x + 2\right)}{2 x^{2} + 4 x + 4} + \operatorname{atan}{\left(x + 1 \right)}$

2. Now simplify:

   $$\frac{x + \left(x^{2} + 2 x + 2\right) \operatorname{atan}{\left(x + 1 \right)} - \frac{1}{2}}{x^{2} + 2 x + 2}$$

3. Add the constant of integration:

   $$\frac{x + \left(x^{2} + 2 x + 2\right) \operatorname{atan}{\left(x + 1 \right)} - \frac{1}{2}}{x^{2} + 2 x + 2}+ \mathrm{constant}$$

The answer is:

$$\frac{x + \left(x^{2} + 2 x + 2\right) \operatorname{atan}{\left(x + 1 \right)} - \frac{1}{2}}{x^{2} + 2 x + 2}+ \mathrm{constant}$$