\int\frac{x^2\;dx}{\sqrt{x^2+a^2}} = \frac{x}{2}\sqrt{x^2+a^2\frac{a^2}{2}\,\mathrm{arcsinh}\frac{x}{a} = \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln\left|x+\sqrt{x^2+a^2}\right|