\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a e \left (e x \right )^{\frac {5}{2}}}-\frac {2 \left (A b +B a \right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a \,e^{3} \sqrt {e x}}+\frac {12 b \left (A b +B a \right ) \left (e x \right )^{\frac {3}{2}} \sqrt {b \,x^{2}+a}}{5 a \,e^{5}}+\frac {24 \left (A b +B a \right ) \sqrt {b}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}{5 e^{4} \left (\sqrt {a}+x \sqrt {b}\right )}-\frac {24 a^{\frac {1}{4}} b^{\frac {1}{4}} \left (A b +B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) e^{\frac {7}{2}} \sqrt {b \,x^{2}+a}}+\frac {12 a^{\frac {1}{4}} b^{\frac {1}{4}} \left (A b +B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) e^{\frac {7}{2}} \sqrt {b \,x^{2}+a}} \]
command
integrate((b*x^2+a)^(3/2)*(B*x^2+A)/(e*x)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (12 \, {\left (B a + A b\right )} \sqrt {b} x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (B b x^{4} - {\left (5 \, B a + 7 \, A b\right )} x^{2} - A a\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {7}{2}\right )}}{5 \, x^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B b x^{4} + {\left (B a + A b\right )} x^{2} + A a\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{e^{4} x^{4}}, x\right ) \]