Answer
\[\frac{2}{3}\,{\left( {3{x^2} + x} \right)^{\frac{3}{2}}} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\,\left( {6x + 1} \right)\sqrt {3{x^2} + x} dx} \hfill \\
\hfill \\
rewrite \hfill \\
\hfill \\
\int_{}^{} {\sqrt {3{x^2} + x} } \,\left( {6x + 1} \right)dx \hfill \\
\hfill \\
set\,\,the\,\,substitution \hfill \\
\hfill \\
u = 3{x^2} + x\,\,\,\,\,\,\,then\,\,\,\,\,du = \,\left( {6x + 1} \right)dx \hfill \\
\hfill \\
apply\,\,the\,\,\,substitution \hfill \\
\hfill \\
\int_{}^{} {\sqrt u du} \hfill \\
\hfill \\
integrate\,\, \hfill \\
\hfill \\
\frac{2}{3}{u^{\frac{3}{2}}} + C \hfill \\
\hfill \\
replace\,\,u\,\,with\,\,\,u = 3{x^2} + x \hfill \\
\hfill \\
\frac{2}{3}\,{\left( {3{x^2} + x} \right)^{\frac{3}{2}}} + C \hfill \\
\end{gathered} \]