The integral $$ F(x) = \int \frac{x}{\sqrt{x^4+10x^2-96x-71}} \, dx ,$$ has a closed-form expression, namely, $$F(x) = -\frac{1}{8} \ln\left((x^6 + 15x^4-80x^3 + 27x^2-528x + 781)\sqrt{x^4 + 10x^2-96x - 71} - (x^8 + 20x^6 -128 x^5 + 54x^4-1408x^3 + 3124x^2 + 10001)\right) \\ +C. $$
It seems that most, if not all, computer integrators such as Mathematica fail to find its antiderivative. What makes this integral so hard to solve, and how do you begin solving it?