$\int \sqrt{x^{2}+5}\,dx$ I tried:

But the answer is: $$\int \sqrt{x^{2}+5}\,dx=\frac{x\sqrt{x^{2}+5}}{2}+\frac{5}{2}\ln(x+\sqrt{x^{2}+5})+c$$
Where is my mistake? I checked my answer several times But still cant find out why my answer is wrong

Your final answer is also correct. We have
\begin{align*}
\dfrac{5}{2}\ln\left|\dfrac{x+\sqrt{x^2+5}}{\sqrt{5}}\right|&=\dfrac{5}{2}\left(\ln|x+\sqrt{x^2+5}|-\ln\sqrt{5}\right)\\&=
\dfrac{5}{2}\ln(x+\sqrt{x^2+5})-\dfrac{5}{2}\ln\sqrt{5}\\&=
\dfrac{5}{2}\ln(x+\sqrt{x^2+5})-5\ln5.
\end{align*}
The absolute-value sign is not necessary since 
$$\forall x\in\mathbb{R}:\;\; x+\sqrt{x^2+5}>0$$
and the constant number $-5\ln 5$ can be absorbed in the contant of integration $C$.
