Solve integral without partial fractions or integration by parts

The following development may seem unmotivated, but it does work. Let $$ v := e^x, \;\; w := 1 + x + v, \;\; w' = 1 + v = w - x. $$ We are trying to integrate $$ u := \frac{x^2+x}{(e^x+x+1)^2} $$ Notice the equalities $$ u = \frac{x (1 + x)}{w^2} = \frac{x (w - v)}{w^2} = \frac{x}{w} - \frac{x v}{w^2} = 1 - \frac{w - x}w - \frac{x v}{w^2} = 1 - \frac{w'}w - \frac{ x v}{w^2}$$ We suppose for some unknown $\,t\,$ $$ y := \frac{t}{w} \;\; \text{ and } \;\; y' = \frac{t'\, w - t\, w'}{w^2} = \frac{t'\, w - t\, (w-x)}{w^2} = \frac{(t'-t)w + t x}{w^2} = -\frac {x v}{w^2}. $$ To solve this last equation, notice that $$ (t' - t)(1 + x + v) + t x = - x v $$ solved by algebra implies that $\,t = 1+x\,$ and $\, t' = 1.\,$ Now $$ \int u\,dx = C + x - \ln(w) + \frac{1+x}{w}. $$

In a situation like this, it helps to have lots of practice with integrals ranging from simple to more complicated ones. It is also very helpful if you already know what the answer is using Computer Algebra Systems and thus you can plot a path towards that goal. You specified

without partial fractions or integration by parts

but these kind of tools are always in the back of the mind. They help to guide the thought process but they do not need to be written down explicitly. You may be able to detect hints of partial fractions and integration by parts even if they are not made explicit.