\[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A}{5 a e \left (e x \right )^{\frac {5}{2}} \sqrt {b \,x^{2}+a}}+\frac {-7 A b +5 B a}{5 a^{2} e^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}+\frac {3 \left (7 A b -5 B a \right ) \sqrt {b \,x^{2}+a}}{5 a^{3} e^{3} \sqrt {e x}}-\frac {3 \left (7 A b -5 B a \right ) \sqrt {b}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}{5 a^{3} e^{4} \left (\sqrt {a}+x \sqrt {b}\right )}+\frac {3 b^{\frac {1}{4}} \left (7 A b -5 B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) a^{\frac {11}{4}} e^{\frac {7}{2}} \sqrt {b \,x^{2}+a}}-\frac {3 b^{\frac {1}{4}} \left (7 A b -5 B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{10 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) a^{\frac {11}{4}} e^{\frac {7}{2}} \sqrt {b \,x^{2}+a}} \]
command
integrate((B*x^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (3 \, {\left ({\left (5 \, B a b - 7 \, A b^{2}\right )} x^{5} + {\left (5 \, B a^{2} - 7 \, A a b\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, {\left (5 \, B a b - 7 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} + 2 \, {\left (5 \, B a^{2} - 7 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {7}{2}\right )}}{5 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{b^{2} e^{4} x^{8} + 2 \, a b e^{4} x^{6} + a^{2} e^{4} x^{4}}, x\right ) \]