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