\[ \int x^{5/2} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx \]
Optimal antiderivative \[ \frac {2 B \,x^{\frac {3}{2}} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{15 c}-\frac {4 b \left (-5 A c +3 b B \right ) x^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{385 c^{2}}-\frac {2 \left (-5 A c +3 b B \right ) x^{\frac {7}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{55 c}+\frac {4 b^{2} \left (-5 A c +3 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{231 c^{3} \sqrt {x}}-\frac {2 b^{\frac {11}{4}} \left (-5 A c +3 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{231 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {13}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(x^(5/2)*(B*x^2+A)*(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
\[ -\frac {2 \, {\left (10 \, {\left (3 \, B b^{4} - 5 \, A b^{3} c\right )} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) - {\left (77 \, B c^{4} x^{6} + 30 \, B b^{3} c - 50 \, A b^{2} c^{2} + 7 \, {\left (2 \, B b c^{3} + 15 \, A c^{4}\right )} x^{4} - 6 \, {\left (3 \, B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{1155 \, c^{4} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B x^{4} + A x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}, x\right ) \]