\[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt {x}} \, dx \]
Optimal antiderivative \[ \frac {2 B \left (c \,x^{4}+b \,x^{2}\right )^{\frac {5}{2}}}{17 c \,x^{\frac {3}{2}}}-\frac {2 \left (-17 A c +7 b B \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \sqrt {x}}{221 c}+\frac {8 b^{3} \left (-17 A c +7 b B \right ) x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{1105 c^{\frac {5}{2}} \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {4 b \left (-17 A c +7 b B \right ) x^{\frac {5}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{663 c}-\frac {8 b^{2} \left (-17 A c +7 b B \right ) \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}}}{3315 c^{2}}-\frac {8 b^{\frac {13}{4}} \left (-17 A c +7 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{1105 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {4 b^{\frac {13}{4}} \left (-17 A c +7 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{1105 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (12 \, {\left (7 \, B b^{4} - 17 \, A b^{3} c\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) - {\left (195 \, B c^{4} x^{6} - 28 \, B b^{3} c + 68 \, A b^{2} c^{2} + 15 \, {\left (19 \, B b c^{3} + 17 \, A c^{4}\right )} x^{4} + 5 \, {\left (4 \, B b^{2} c^{2} + 85 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{3315 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B c x^{5} + {\left (B b + A c\right )} x^{3} + A b x\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}, x\right ) \]