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