\[ \int \frac {1}{\sqrt {x} \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{b \,x^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {21 c^{\frac {3}{2}} x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{5 b^{3} \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {7 \sqrt {c \,x^{4}+b \,x^{2}}}{5 b^{2} x^{\frac {7}{2}}}+\frac {21 c \sqrt {c \,x^{4}+b \,x^{2}}}{5 b^{3} x^{\frac {3}{2}}}+\frac {21 c^{\frac {5}{4}} x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {11}{4}} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {21 c^{\frac {5}{4}} x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{10 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {11}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(1/(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
\[ \frac {21 \, {\left (c^{2} x^{6} + b c x^{4}\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + {\left (21 \, c^{2} x^{4} + 14 \, b c x^{2} - 2 \, b^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}}{5 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {x}}{c^{2} x^{9} + 2 \, b c x^{7} + b^{2} x^{5}}, x\right ) \]