\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (9 A b +B a \right ) \left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 a \,e^{3}}-\frac {2 A \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a e \sqrt {e x}}+\frac {4 \left (9 A b +B a \right ) \left (e x \right )^{\frac {3}{2}} \sqrt {b \,x^{2}+a}}{15 e^{3}}+\frac {8 a \left (9 A b +B a \right ) \sqrt {e x}\, \sqrt {b \,x^{2}+a}}{15 e^{2} \sqrt {b}\, \left (\sqrt {a}+x \sqrt {b}\right )}-\frac {8 a^{\frac {5}{4}} \left (9 A b +B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) b^{\frac {3}{4}} e^{\frac {3}{2}} \sqrt {b \,x^{2}+a}}+\frac {4 a^{\frac {5}{4}} \left (9 A b +B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) b^{\frac {3}{4}} e^{\frac {3}{2}} \sqrt {b \,x^{2}+a}} \]
command
integrate((b*x^2+a)^(3/2)*(B*x^2+A)/(e*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (12 \, {\left (B a^{2} + 9 \, A a b\right )} \sqrt {b} x {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (5 \, B b^{2} x^{4} - 45 \, A a b + {\left (11 \, B a b + 9 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{45 \, b x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B b x^{4} + {\left (B a + A b\right )} x^{2} + A a\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{e^{2} x^{2}}, x\right ) \]