\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a e \left (e x \right )^{\frac {3}{2}}}+\frac {2 \left (7 A b +3 B a \right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x}}{21 a \,e^{3}}+\frac {4 \left (7 A b +3 B a \right ) \sqrt {e x}\, \sqrt {b \,x^{2}+a}}{21 e^{3}}+\frac {4 a^{\frac {3}{4}} \left (7 A b +3 B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{21 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) b^{\frac {1}{4}} e^{\frac {5}{2}} \sqrt {b \,x^{2}+a}} \]
command
integrate((b*x^2+a)^(3/2)*(B*x^2+A)/(e*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (4 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} \sqrt {b} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (3 \, B b^{2} x^{4} - 7 \, A a b + {\left (9 \, B a b + 7 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {5}{2}\right )}}{21 \, b x^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B b x^{4} + {\left (B a + A b\right )} x^{2} + A a\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{e^{3} x^{3}}, x\right ) \]