\[ \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx \]
Optimal antiderivative \[ \frac {\left (6 A b -a B \right ) \left (e x \right )^{\frac {3}{2}} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{36 b e}+\frac {B \left (e x \right )^{\frac {3}{2}} \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{9 b e}+\frac {a^{2} \left (6 A b -a B \right ) \arctanh \left (\frac {\left (e x \right )^{\frac {3}{2}} \sqrt {b}}{e^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}\right ) \sqrt {e}}{24 b^{\frac {3}{2}}}+\frac {a \left (6 A b -a B \right ) \left (e x \right )^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}{24 b e} \]
command
integrate((b*x^3+a)^(3/2)*(B*x^3+A)*(e*x)^(1/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {1}{144} \, {\left (6 \, {\left (\frac {3 \, a^{2} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{\sqrt {b}} + \frac {2 \, {\left (\frac {3 \, \sqrt {b x^{3} + a} a^{2} b}{x^{\frac {3}{2}}} - \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {9}{2}}}\right )}}{b^{2} - \frac {2 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2}}{x^{6}}}\right )} A - {\left (\frac {3 \, a^{3} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, {\left (\frac {3 \, \sqrt {b x^{3} + a} a^{3} b^{2}}{x^{\frac {3}{2}}} - \frac {8 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {9}{2}}} - \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {15}{2}}}\right )}}{b^{4} - \frac {3 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {3 \, {\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac {{\left (b x^{3} + a\right )}^{3} b}{x^{9}}}\right )} B\right )} e^{\frac {1}{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \sqrt {e x}\,{d x} \]________________________________________________________________________________________