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$$\phantom=\int\frac{x^3+\green{7x^2}+\red{15x}+\blue{10}}{x^2+6x+9}\,dx$$$$=\int\frac{x^3+\green{6x^2}+\green{x^2}+\red{9x+6x}+\blue{9+1}}{x^2+6x+9}\,dx$$$$=\int\frac{(x^3+\green{6x^2}+\red{9x})+(\green{x^2}+\red{6x}+\blue9)+\blue1}{x^2+6x+9}\,dx$$$$=\int\left(\frac{x(x^2+6x+9)}{x^2+6x+9}+\frac{x^2+6x+9}{x^2+6x+9}+\frac{1}{x^2+6x+9}\right)dx$$$$=\int\left(x+1+\frac{1}{(x+3)^2}\right)dx$$$$=\frac{x^2}{2}+x-\frac{1}{x+3}+\text{const}$$