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