Answer
\[ = \ln \left| {x + 5 + \sqrt {\,{{\left( {x + 5} \right)}^2} - 25} } \right| + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\frac{{dx}}{{\sqrt {{x^2} + 10x} }}} \hfill \\
\hfill \\
se\,t\,\,\,\,\,x + 5 = u\,\,\,\,then\,\,\,\,dx = du \hfill \\
\hfill \\
\int_{}^{} {\frac{{dx}}{{\sqrt {{x^2} + 10x} }}} = \int_{}^{} {\frac{{dx}}{{\sqrt {{u^2} - 5} }}} \hfill \\
\hfill \\
{\text{using}}\,\,the\,\,formula\,\, \hfill \\
\hfill \\
\int_{}^{} {\frac{{dx}}{{\sqrt {{x^2} - {a^2}} }}} = \ln \left| {x + \sqrt {{x^2} - {a^2}} } \right| + C \hfill \\
\hfill \\
then \hfill \\
\hfill \\
\int_{}^{} {\frac{{du}}{{\sqrt {{u^2} - 5} }}} = \ln \left| {u + \sqrt {{u^2} - {5^2}} } \right| + C \hfill \\
\hfill \\
= \ln \left| {u + \sqrt {{u^2} - 25} } \right| + C \hfill \\
\hfill \\
substitute\,\,back\,\,u = x + 5 \hfill \\
\hfill \\
= \ln \left| {x + 5 + \sqrt {\,{{\left( {x + 5} \right)}^2} - 25} } \right| + C \hfill \\
\hfill \\
\end{gathered} \]