Answer
$$\frac{{{5^{{{\left( {x + 1} \right)}^2}}}}}{{2\ln 5}} + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {x + 1} \right){5^{{{\left( {x + 1} \right)}^2}}}dx} \cr
& {\text{Integrate using the substitution method}} \cr
& {\text{Let }}u = {\left( {x + 1} \right)^2},{\text{ }}du = 2\left( {x + 1} \right)dx,{\text{ }}dx = \frac{1}{{2\left( {x + 1} \right)}}du \cr
& {\text{Substituting}} \cr
& \int {\left( {x + 1} \right){5^{{{\left( {x + 1} \right)}^2}}}dx} = \int {\left( {x + 1} \right){5^u}\frac{1}{{2\left( {x + 1} \right)}}du} \cr
& = \frac{1}{2}\int {{5^u}du} \cr
& {\text{ = }}\frac{1}{2}\left( {\frac{{{5^u}}}{{\ln 5}}} \right) + C \cr
& {\text{ = }}\frac{{{5^u}}}{{2\ln 5}} + C \cr
& {\text{Write in terms of }}x \cr
& {\text{ = }}\frac{{{5^{{{\left( {x + 1} \right)}^2}}}}}{{2\ln 5}} + C \cr} $$
