\[ \int \frac {\sqrt {e x} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (A b -B a \right ) \left (e x \right )^{\frac {3}{2}}}{9 a b e \left (b \,x^{3}+a \right )^{\frac {3}{2}}}+\frac {2 \left (2 A b +B a \right ) \left (e x \right )^{\frac {3}{2}}}{9 a^{2} b e \sqrt {b \,x^{3}+a}} \]
command
integrate((B*x**3+A)*(e*x)**(1/2)/(b*x**3+a)**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ A \left (\frac {6 a \sqrt {e} x^{\frac {3}{2}}}{9 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{3}}{a}} + 9 a^{\frac {5}{2}} b x^{3} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {4 b \sqrt {e} x^{\frac {9}{2}}}{9 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{3}}{a}} + 9 a^{\frac {5}{2}} b x^{3} \sqrt {1 + \frac {b x^{3}}{a}}}\right ) + \frac {2 B \sqrt {e} x^{\frac {9}{2}}}{9 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}} + 9 a^{\frac {3}{2}} b x^{3} \sqrt {1 + \frac {b x^{3}}{a}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________