\[ \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-5 A c +3 b B \right ) x^{\frac {9}{2}} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{105 c}+\frac {2 B \,x^{\frac {5}{2}} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {5}{2}}}{25 c}+\frac {88 b^{5} \left (-5 A c +3 b B \right ) x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{16575 c^{\frac {9}{2}} \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {88 b^{3} \left (-5 A c +3 b B \right ) x^{\frac {5}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{69615 c^{3}}-\frac {8 b^{2} \left (-5 A c +3 b B \right ) x^{\frac {9}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{7735 c^{2}}-\frac {4 b \left (-5 A c +3 b B \right ) x^{\frac {13}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{595 c}-\frac {88 b^{4} \left (-5 A c +3 b B \right ) \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}}}{49725 c^{4}}-\frac {88 b^{\frac {21}{4}} \left (-5 A c +3 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{16575 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {19}{4}} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {44 b^{\frac {21}{4}} \left (-5 A c +3 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{16575 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {19}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(x^(7/2)*(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
\[ -\frac {2 \, {\left (924 \, {\left (3 \, B b^{6} - 5 \, A b^{5} c\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) - {\left (13923 \, B c^{6} x^{10} + 663 \, {\left (27 \, B b c^{5} + 25 \, A c^{6}\right )} x^{8} - 924 \, B b^{5} c + 1540 \, A b^{4} c^{2} + 39 \, {\left (12 \, B b^{2} c^{4} + 575 \, A b c^{5}\right )} x^{6} - 180 \, {\left (3 \, B b^{3} c^{3} - 5 \, A b^{2} c^{4}\right )} x^{4} + 220 \, {\left (3 \, B b^{4} c^{2} - 5 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{348075 \, c^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B c x^{9} + {\left (B b + A c\right )} x^{7} + A b x^{5}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}, x\right ) \]