\[ \int \frac {(e x)^{7/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (A b -a B \right ) \left (e x \right )^{\frac {9}{2}}}{9 a b e \left (b \,x^{3}+a \right )^{\frac {3}{2}}}+\frac {2 B \,e^{\frac {7}{2}} \arctanh \left (\frac {\left (e x \right )^{\frac {3}{2}} \sqrt {b}}{e^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}\right )}{3 b^{\frac {5}{2}}}-\frac {2 B \,e^{2} \left (e x \right )^{\frac {3}{2}}}{3 b^{2} \sqrt {b \,x^{3}+a}} \]
command
integrate((e*x)^(7/2)*(B*x^3+A)/(b*x^3+a)^(5/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{9} \, {\left (\frac {2 \, A x^{\frac {9}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a} - {\left (\frac {2 \, {\left (b + \frac {3 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{\frac {9}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2}} + \frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{b^{\frac {5}{2}}}\right )} B\right )} e^{\frac {7}{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________