\[ \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-21 A c +11 b B \right ) x^{\frac {5}{2}} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{357 c}+\frac {2 B \left (c \,x^{4}+b \,x^{2}\right )^{\frac {5}{2}} \sqrt {x}}{21 c}-\frac {8 b^{4} \left (-21 A c +11 b B \right ) x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{3315 c^{\frac {7}{2}} \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {8 b^{2} \left (-21 A c +11 b B \right ) x^{\frac {5}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{13923 c^{2}}-\frac {4 b \left (-21 A c +11 b B \right ) x^{\frac {9}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{1547 c}+\frac {8 b^{3} \left (-21 A c +11 b B \right ) \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}}}{9945 c^{3}}+\frac {8 b^{\frac {17}{4}} \left (-21 A c +11 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{3315 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {15}{4}} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {4 b^{\frac {17}{4}} \left (-21 A c +11 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{3315 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {15}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(x^(3/2)*(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
\[ \frac {2 \, {\left (84 \, {\left (11 \, B b^{5} - 21 \, A b^{4} c\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + {\left (3315 \, B c^{5} x^{8} + 195 \, {\left (23 \, B b c^{4} + 21 \, A c^{5}\right )} x^{6} + 308 \, B b^{4} c - 588 \, A b^{3} c^{2} + 45 \, {\left (4 \, B b^{2} c^{3} + 133 \, A b c^{4}\right )} x^{4} - 20 \, {\left (11 \, B b^{3} c^{2} - 21 \, A b^{2} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{69615 \, c^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B c x^{7} + {\left (B b + A c\right )} x^{5} + A b x^{3}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}, x\right ) \]