\[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (7 A b -9 B a \right ) e \left (e x \right )^{\frac {5}{2}}}{7 b^{2} \sqrt {b \,x^{2}+a}}+\frac {2 B \left (e x \right )^{\frac {9}{2}}}{7 b e \sqrt {b \,x^{2}+a}}+\frac {5 \left (7 A b -9 B a \right ) e^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}{21 b^{3}}-\frac {5 a^{\frac {3}{4}} \left (7 A b -9 B a \right ) e^{\frac {7}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{42 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) b^{\frac {13}{4}} \sqrt {b \,x^{2}+a}} \]
command
integrate((e*x)^(7/2)*(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
\[ \frac {5 \, {\left (9 \, B a^{3} - 7 \, A a^{2} b + {\left (9 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (6 \, B b^{3} x^{4} - 45 \, B a^{2} b + 35 \, A a b^{2} - 2 \, {\left (9 \, B a b^{2} - 7 \, A b^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x} e^{\frac {7}{2}}}{21 \, {\left (b^{5} x^{2} + a b^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B e^{3} x^{5} + A e^{3} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]