\[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (7 A b -B a \right ) \left (e x \right )^{\frac {3}{2}}}{3 a^{2} e^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 A}{a e \left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x}}-\frac {\left (7 A b -B a \right ) \left (e x \right )^{\frac {3}{2}}}{2 a^{3} e^{3} \sqrt {b \,x^{2}+a}}+\frac {\left (7 A b -B a \right ) \sqrt {e x}\, \sqrt {b \,x^{2}+a}}{2 a^{3} e^{2} \sqrt {b}\, \left (\sqrt {a}+x \sqrt {b}\right )}-\frac {\left (7 A b -B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{2 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) a^{\frac {11}{4}} b^{\frac {3}{4}} e^{\frac {3}{2}} \sqrt {b \,x^{2}+a}}+\frac {\left (7 A b -B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{4 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) a^{\frac {11}{4}} b^{\frac {3}{4}} e^{\frac {3}{2}} \sqrt {b \,x^{2}+a}} \]
command
integrate((B*x**2+A)/(e*x)**(3/2)/(b*x**2+a)**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________