\[ \int x^{5/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-23 A c +13 b B \right ) x^{\frac {7}{2}} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{437 c}+\frac {2 B \,x^{\frac {3}{2}} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {5}{2}}}{23 c}+\frac {72 b^{3} \left (-23 A c +13 b B \right ) x^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{168245 c^{3}}-\frac {8 b^{2} \left (-23 A c +13 b B \right ) x^{\frac {7}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{24035 c^{2}}-\frac {4 b \left (-23 A c +13 b B \right ) x^{\frac {11}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{2185 c}-\frac {24 b^{4} \left (-23 A c +13 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{33649 c^{4} \sqrt {x}}+\frac {12 b^{\frac {19}{4}} \left (-23 A c +13 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{33649 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {17}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(x^(5/2)*(B*x^2+A)*(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
\[ \frac {2 \, {\left (60 \, {\left (13 \, B b^{6} - 23 \, A b^{5} c\right )} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (7315 \, B c^{6} x^{10} + 385 \, {\left (25 \, B b c^{5} + 23 \, A c^{6}\right )} x^{8} - 780 \, B b^{5} c + 1380 \, A b^{4} c^{2} + 77 \, {\left (4 \, B b^{2} c^{4} + 161 \, A b c^{5}\right )} x^{6} - 28 \, {\left (13 \, B b^{3} c^{3} - 23 \, A b^{2} c^{4}\right )} x^{4} + 36 \, {\left (13 \, B b^{4} c^{2} - 23 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{168245 \, c^{5} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B c x^{8} + {\left (B b + A c\right )} x^{6} + A b x^{4}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}, x\right ) \]