\[ \int \frac {A+B x^2}{x^{7/2} \sqrt {b x^2+c x^4}} \, dx \]
Optimal antiderivative \[ -\frac {2 A \sqrt {c \,x^{4}+b \,x^{2}}}{7 b \,x^{\frac {9}{2}}}-\frac {2 \left (-5 A c +7 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{21 b^{2} x^{\frac {5}{2}}}-\frac {c^{\frac {3}{4}} \left (-5 A c +7 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{21 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {9}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((B*x^2+A)/x^(7/2)/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (7 \, B b - 5 \, A c\right )} \sqrt {c} x^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + \sqrt {c x^{4} + b x^{2}} {\left ({\left (7 \, B b - 5 \, A c\right )} x^{2} + 3 \, A b\right )} \sqrt {x}\right )}}{21 \, b^{2} x^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x}}{c x^{8} + b x^{6}}, x\right ) \]