Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 275: 31

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

Given $$\int \frac{x+1}{\left(x^{2}+2 x\right)^{3}} d x$$ Let $$u=x^{2}+2 x\ \ \ \Rightarrow \ \ du=2(x+1)dx $$ then \begin{align*} \int \frac{x+1}{\left(x^{2}+2 x\right)^{3}} d x&=\frac{1}{2}\int \frac{du}{u^{3}} \\ &=\frac{-1}{4u^2}+c\\ &=\frac{-1}{4 \left(x^{2}+2 x\right)^2}+c\\ \end{align*}