\[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A}{3 a e \left (e x \right )^{\frac {3}{2}} \sqrt {b \,x^{2}+a}}-\frac {\left (5 A b -3 B a \right ) \sqrt {e x}}{3 a^{2} e^{3} \sqrt {b \,x^{2}+a}}-\frac {\left (5 A b -3 B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{6 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) a^{\frac {9}{4}} b^{\frac {1}{4}} e^{\frac {5}{2}} \sqrt {b \,x^{2}+a}} \]
command
integrate((B*x^2+A)/(e*x)^(5/2)/(b*x^2+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left ({\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{4} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (2 \, A a b - {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {5}{2}\right )}}{3 \, {\left (a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{b^{2} e^{3} x^{7} + 2 \, a b e^{3} x^{5} + a^{2} e^{3} x^{3}}, x\right ) \]