\[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{15/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (3 A c +7 b B \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{21 b \,x^{\frac {9}{2}}}-\frac {2 A \left (c \,x^{4}+b \,x^{2}\right )^{\frac {5}{2}}}{7 b \,x^{\frac {17}{2}}}+\frac {4 c \left (3 A c +7 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{21 b \sqrt {x}}+\frac {4 c^{\frac {3}{4}} \left (3 A c +7 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{21 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {1}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(15/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (4 \, {\left (7 \, B b + 3 \, A c\right )} \sqrt {c} x^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (7 \, B c x^{4} - {\left (7 \, B b + 9 \, A c\right )} x^{2} - 3 \, A b\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{21 \, x^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B c x^{4} + {\left (B b + A c\right )} x^{2} + A b\right )} \sqrt {c x^{4} + b x^{2}}}{x^{\frac {11}{2}}}, x\right ) \]