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