\[ \int \frac {\sqrt {e x} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (A b -a B \right ) \left (e x \right )^{\frac {3}{2}}}{9 a b e \left (b \,x^{3}+a \right )^{\frac {3}{2}}}+\frac {2 \left (2 A b +a B \right ) \left (e x \right )^{\frac {3}{2}}}{9 a^{2} b e \sqrt {b \,x^{3}+a}} \]
command
integrate((B*x^3+A)*(e*x)^(1/2)/(b*x^3+a)^(5/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {2}{9} \, {\left (\frac {B x^{\frac {9}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a} - \frac {A {\left (b - \frac {3 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{\frac {9}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}\right )} e^{\frac {1}{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\left (B x^{3} + A\right )} \sqrt {e x}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________