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