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