\[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A}{9 b \,x^{\frac {7}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {-11 A c +9 b B}{9 b^{2} x^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {7 c^{\frac {3}{2}} \left (-11 A c +9 b B \right ) x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{15 b^{4} \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {7 \left (-11 A c +9 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{45 b^{3} x^{\frac {7}{2}}}+\frac {7 c \left (-11 A c +9 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{15 b^{4} x^{\frac {3}{2}}}+\frac {7 c^{\frac {5}{4}} \left (-11 A c +9 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {15}{4}} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {7 c^{\frac {5}{4}} \left (-11 A c +9 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{30 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {15}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((B*x^2+A)/x^(5/2)/(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {21 \, {\left ({\left (9 \, B b c^{2} - 11 \, A c^{3}\right )} x^{8} + {\left (9 \, B b^{2} c - 11 \, A b c^{2}\right )} x^{6}\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + {\left (21 \, {\left (9 \, B b c^{2} - 11 \, A c^{3}\right )} x^{6} + 14 \, {\left (9 \, B b^{2} c - 11 \, A b c^{2}\right )} x^{4} - 10 \, A b^{3} - 2 \, {\left (9 \, B b^{3} - 11 \, A b^{2} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}}{45 \, {\left (b^{4} c x^{8} + b^{5} x^{6}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x}}{c^{2} x^{11} + 2 \, b c x^{9} + b^{2} x^{7}}, x\right ) \]