\[ \int \frac {A+B x^3}{(e x)^{5/2} \left (a+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A}{3 a e \left (e x \right )^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}-\frac {2 \left (2 A b -a B \right ) \left (e x \right )^{\frac {3}{2}}}{3 a^{2} e^{4} \sqrt {b \,x^{3}+a}} \]
command
integrate((B*x^3+A)/(e*x)^(5/2)/(b*x^3+a)^(3/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {2}{3} \, {\left (A {\left (\frac {b x^{\frac {3}{2}}}{\sqrt {b x^{3} + a} a^{2}} + \frac {\sqrt {b x^{3} + a}}{a^{2} x^{\frac {3}{2}}}\right )} - \frac {B x^{\frac {3}{2}}}{\sqrt {b x^{3} + a} a}\right )} e^{\left (-\frac {5}{2}\right )} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________