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