Calculate $\int_0^\infty {\frac{x}{{\left( {x + 1} \right)\sqrt {4{x^4} + 8{x^3} + 12{x^2} + 8x + 1} }}dx}$

This is a pseudo-elliptic integral, it has an elementary anti-derivative:

$$\int \frac{x}{(x+1)\sqrt{4x^4+8x^3+12x^2+8x+1}} dx = \frac{\ln\left[P(x)+Q(x)\sqrt{4x^4+8x^3+12x^2+8x+1}\right]}{6} - \ln(x+1) + C$$

where $$P(x) = 112x^6+360x^5+624x^4+772x^3+612x^2+258x+43$$ and $$Q(x) = 52x^4+92x^3+30x^2-22x-11$$

To obtain this answer, just follow the systematic method of symbolic integration over simple algebraic extension. Alternatively, you can throw it to a CAS with Risch algorithm implemented (not Mathematica), a convenient software is the online Axiom sandbox.