\[ \int x^{3/2} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx \]
Optimal antiderivative \[ \frac {2 B \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \sqrt {x}}{13 c}+\frac {4 b^{2} \left (-13 A c +7 b B \right ) x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{195 c^{\frac {5}{2}} \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {2 \left (-13 A c +7 b B \right ) x^{\frac {5}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{117 c}-\frac {4 b \left (-13 A c +7 b B \right ) \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}}}{585 c^{2}}-\frac {4 b^{\frac {9}{4}} \left (-13 A c +7 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{195 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {2 b^{\frac {9}{4}} \left (-13 A c +7 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{195 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(x^(3/2)*(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
\[ -\frac {2 \, {\left (6 \, {\left (7 \, B b^{3} - 13 \, A b^{2} c\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) - {\left (45 \, B c^{3} x^{4} - 14 \, B b^{2} c + 26 \, A b c^{2} + 5 \, {\left (2 \, B b c^{2} + 13 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{585 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {c x^{4} + b x^{2}} {\left (B x^{3} + A x\right )} \sqrt {x}, x\right ) \]