\[ \int \frac {(e x)^{7/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (2 A b -3 a B \right ) e^{\frac {7}{2}} \arctanh \left (\frac {\left (e x \right )^{\frac {3}{2}} \sqrt {b}}{e^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}\right )}{3 b^{\frac {5}{2}}}-\frac {\left (2 A b -3 a B \right ) e^{2} \left (e x \right )^{\frac {3}{2}}}{3 b^{2} \sqrt {b \,x^{3}+a}}+\frac {B \left (e x \right )^{\frac {9}{2}}}{3 b e \sqrt {b \,x^{3}+a}} \]
command
integrate((e*x)^(7/2)*(B*x^3+A)/(b*x^3+a)^(3/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{6} \, {\left (B {\left (\frac {2 \, {\left (2 \, a b - \frac {3 \, {\left (b x^{3} + a\right )} a}{x^{3}}\right )}}{\frac {\sqrt {b x^{3} + a} b^{3}}{x^{\frac {3}{2}}} - \frac {{\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {9}{2}}}} + \frac {3 \, a \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{b^{\frac {5}{2}}}\right )} - 2 \, A {\left (\frac {2 \, x^{\frac {3}{2}}}{\sqrt {b x^{3} + a} b} + \frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{b^{\frac {3}{2}}}\right )}\right )} e^{\frac {7}{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________