\[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{23/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{15 x^{\frac {21}{2}}}-\frac {4 c \sqrt {c \,x^{4}+b \,x^{2}}}{55 x^{\frac {13}{2}}}-\frac {8 c^{2} \sqrt {c \,x^{4}+b \,x^{2}}}{385 b \,x^{\frac {9}{2}}}+\frac {8 c^{3} \sqrt {c \,x^{4}+b \,x^{2}}}{231 b^{2} x^{\frac {5}{2}}}+\frac {4 c^{\frac {15}{4}} x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{231 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {9}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((c*x^4+b*x^2)^(3/2)/x^(23/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (20 \, c^{\frac {7}{2}} x^{9} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (20 \, c^{3} x^{6} - 12 \, b c^{2} x^{4} - 119 \, b^{2} c x^{2} - 77 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{1155 \, b^{2} x^{9}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + b\right )}}{x^{\frac {19}{2}}}, x\right ) \]