\[ \int \frac {1}{x^{3/2} \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{b \,x^{\frac {5}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {9 \sqrt {c \,x^{4}+b \,x^{2}}}{7 b^{2} x^{\frac {9}{2}}}+\frac {15 c \sqrt {c \,x^{4}+b \,x^{2}}}{7 b^{3} x^{\frac {5}{2}}}+\frac {15 c^{\frac {7}{4}} x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{14 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {13}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(1/x^(3/2)/(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
\[ \frac {15 \, {\left (c^{2} x^{7} + b c x^{5}\right )} \sqrt {c} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (15 \, c^{2} x^{4} + 6 \, b c x^{2} - 2 \, b^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}}{7 \, {\left (b^{3} c x^{7} + b^{4} x^{5}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {x}}{c^{2} x^{10} + 2 \, b c x^{8} + b^{2} x^{6}}, x\right ) \]