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