$\displaystyle \int\limits_0^1\frac{dx}{(1+x)\sqrt{8x^2+14x+5}}$
with substitution
Try rewriting the square root in the denominator with $\displaystyle (x+1)$.
$\displaystyle 8x^2+14x+5 = 8(x+1)^2-2(x+1)-1$
Then use $\displaystyle u=x+1$ and $\displaystyle du=dx$
Now you have $\displaystyle \int_0^1\frac{1}{u\sqrt{8u^2-2u-1}}\,du$
This should make it a little easier, though to be honest I'm not sure where to go from here. I'll edit in additional steps if I think of them.
If you want it, the final answer (according to Maple) is:
Spoiler: