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