\[ \int \sqrt {x} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-19 A c +9 b B \right ) x^{\frac {3}{2}} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{285 c}+\frac {2 B \left (c \,x^{4}+b \,x^{2}\right )^{\frac {5}{2}}}{19 c \sqrt {x}}-\frac {8 b^{2} \left (-19 A c +9 b B \right ) x^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{7315 c^{2}}-\frac {4 b \left (-19 A c +9 b B \right ) x^{\frac {7}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}{1045 c}+\frac {8 b^{3} \left (-19 A c +9 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{4389 c^{3} \sqrt {x}}-\frac {4 b^{\frac {15}{4}} \left (-19 A c +9 b B \right ) x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{4389 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) c^{\frac {13}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
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
\[ -\frac {2 \, {\left (20 \, {\left (9 \, B b^{5} - 19 \, A b^{4} c\right )} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) - {\left (1155 \, B c^{5} x^{8} + 77 \, {\left (21 \, B b c^{4} + 19 \, A c^{5}\right )} x^{6} + 180 \, B b^{4} c - 380 \, A b^{3} c^{2} + 7 \, {\left (12 \, B b^{2} c^{3} + 323 \, A b c^{4}\right )} x^{4} - 12 \, {\left (9 \, B b^{3} c^{2} - 19 \, A b^{2} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{21945 \, c^{4} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B c x^{6} + {\left (B b + A c\right )} x^{4} + A b x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}, x\right ) \]