Answer
$\frac{2}{3}\sqrt {3ax + b{x^3}} + C$
Work Step by Step
$$\eqalign{
& \int {\frac{{a + b{x^2}}}{{\sqrt {3ax + b{x^3}} }}} dx \cr
& {\text{Let }}u = 3ax + b{x^3},{\text{ then }}du = \left( {3a + 3b{x^2}} \right)dx \cr
& du = 3\left( {a + b{x^2}} \right)dx \cr
& \frac{1}{3}du = \left( {a + b{x^2}} \right)dx \cr
& {\text{Applying the substitution}}{\text{, we obtain}} \cr
& \int {\frac{{a + b{x^2}}}{{\sqrt {3ax + b{x^3}} }}} dx = \int {\frac{1}{{\sqrt u }}\left( {\frac{1}{3}} \right)} du \cr
& = \frac{1}{3}\int {\frac{1}{{\sqrt u }}} du \cr
& = \frac{1}{3}\int {{u^{ - 1/2}}} du \cr
& {\text{Integrate}} \cr
& = \frac{1}{3}\left( {\frac{{{u^{1/2}}}}{{1/2}}} \right) + C \cr
& = \frac{2}{3}{u^{1/2}} + C \cr
& = \frac{2}{3}\sqrt u + C \cr
& {\text{Write in terms of }}x,{\text{ substitute }}u = 3ax + b{x^3} \cr
& = \frac{2}{3}\sqrt {3ax + b{x^3}} + C \cr} $$