Answer
$$\frac{2}{{15}}{\left( {5\ln x + 3} \right)^{3/2}} + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{\sqrt {5\ln x + 3} }}{x}} dx \cr
& {\text{set }}u = 5\ln x + 3{\text{ then }}\frac{{du}}{{dx}} = \frac{5}{x},\,\,\,\,\,\,\frac{1}{5}du = \frac{1}{x}dx \cr
& {\text{write the integrand in terms of }}u \cr
& \int {\frac{{\sqrt {5\ln x + 3} }}{x}} dx = \int {\sqrt u } \left( {\frac{1}{5}du} \right) \cr
& = \frac{1}{5}\int {\sqrt u } du \cr
& {\text{rewrite the radicand}} \cr
& = \frac{1}{5}\int {{u^{1/2}}} du \cr
& {\text{integrate by using the power rule }}\int {{u^n}} du = \frac{{{u^{n + 1}}}}{{n + 1}} + C \cr
& = \frac{1}{5}\left( {\frac{{{u^{3/2}}}}{{3/2}}} \right) + C \cr
& = \frac{2}{{15}}{u^{3/2}} + C \cr
& {\text{replace }}5\ln x + 3{\text{ for }}u \cr
& = \frac{2}{{15}}{\left( {5\ln x + 3} \right)^{3/2}} + C \cr} $$
