\[ \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx \]
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="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{72} \, {\left (6 \, \sqrt {b x^{3} + a} {\left (2 \, x^{3} + \frac {a}{b}\right )} B a x^{\frac {3}{2}} + 6 \, \sqrt {b x^{3} + a} {\left (2 \, x^{3} + \frac {a}{b}\right )} A b x^{\frac {3}{2}} + {\left (2 \, {\left (4 \, x^{3} + \frac {a}{b}\right )} x^{3} - \frac {3 \, a^{2}}{b^{2}}\right )} \sqrt {b x^{3} + a} B b x^{\frac {3}{2}} + 24 \, {\left (\sqrt {b x^{3} + a} x^{\frac {3}{2}} - \frac {a \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} + \sqrt {b x^{3} + a} \right |}\right )}{\sqrt {b}}\right )} A a\right )} e^{\frac {1}{2}} - \frac {{\left (B^{2} a^{6} + 4 \, A B a^{5} b + 4 \, A^{2} a^{4} b^{2}\right )} e^{\frac {1}{2}} \log \left ({\left | {\left (B a^{3} x^{\frac {3}{2}} + 2 \, A a^{2} b x^{\frac {3}{2}}\right )} \sqrt {b} + \sqrt {B^{2} a^{7} + 4 \, A B a^{6} b + 4 \, A^{2} a^{5} b^{2} + {\left (B a^{3} x^{\frac {3}{2}} + 2 \, A a^{2} b x^{\frac {3}{2}}\right )}^{2} b} \right |}\right )}{24 \, b^{\frac {3}{2}} {\left | B a^{3} + 2 \, A a^{2} b \right |}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________