\[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (A b -B a \right ) \left (e x \right )^{\frac {5}{2}}}{3 a b e \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {\left (A b +5 B a \right ) e \sqrt {e x}}{6 a \,b^{2} \sqrt {b \,x^{2}+a}}+\frac {\left (A b +5 B a \right ) e^{\frac {3}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{12 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) a^{\frac {5}{4}} b^{\frac {9}{4}} \sqrt {b \,x^{2}+a}} \]
command
integrate((e*x)**(3/2)*(B*x**2+A)/(b*x**2+a)**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {A e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {B e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{4}, \frac {5}{2} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {13}{4}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________