Answer
$$ \displaystyle\int \dfrac{{x}^{2}}{\sqrt{{a}^{2}+{x}^{2}}}\, \mathrm d x = -{{a^2\,{\rm asinh}\; \left({{x}\over{\left| a\right| }}\right)-x\, \sqrt{x^2+a^2}}\over{2}} $$
Explanation
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