\[ \int \frac {1}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{b \,x^{\frac {7}{2}} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {77 c^{\frac {5}{2}} x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{15 b^{4} \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {11 \sqrt {c \,x^{4}+b \,x^{2}}}{9 b^{2} x^{\frac {11}{2}}}+\frac {77 c \sqrt {c \,x^{4}+b \,x^{2}}}{45 b^{3} x^{\frac {7}{2}}}-\frac {77 c^{2} \sqrt {c \,x^{4}+b \,x^{2}}}{15 b^{4} x^{\frac {3}{2}}}-\frac {77 c^{\frac {9}{4}} x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {15}{4}} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {77 c^{\frac {9}{4}} x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{30 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {15}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(1/x^(5/2)/(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
\[ -\frac {231 \, {\left (c^{3} x^{8} + b c^{2} x^{6}\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + {\left (231 \, c^{3} x^{6} + 154 \, b c^{2} x^{4} - 22 \, b^{2} c x^{2} + 10 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}}{45 \, {\left (b^{4} c x^{8} + b^{5} x^{6}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {x}}{c^{2} x^{11} + 2 \, b c x^{9} + b^{2} x^{7}}, x\right ) \]