\[ \int \frac {A+B x^3}{(e x)^{5/2} \left (a+b x^3\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A}{3 a e \left (e x \right )^{\frac {3}{2}} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}-\frac {2 \left (4 A b -a B \right ) \left (e x \right )^{\frac {3}{2}}}{9 a^{2} e^{4} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}-\frac {4 \left (4 A b -a B \right ) \left (e x \right )^{\frac {3}{2}}}{9 a^{3} e^{4} \sqrt {b \,x^{3}+a}} \]
command
integrate((B*x^3+A)/(e*x)^(5/2)/(b*x^3+a)^(5/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {2}{9} \, {\left (\frac {B {\left (b - \frac {3 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{\frac {9}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}} - A {\left (\frac {{\left (b^{2} - \frac {6 \, {\left (b x^{3} + a\right )} b}{x^{3}}\right )} x^{\frac {9}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {3 \, \sqrt {b x^{3} + a}}{a^{3} x^{\frac {3}{2}}}\right )}\right )} e^{\left (-\frac {5}{2}\right )} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________