\[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^5} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{4 x^{4}}-\frac {3 b \,c^{2} \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{8 a \sqrt {c \,x^{2}}}+\frac {3 b^{\frac {4}{3}} c^{2} \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{8 a \left (a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )}+\frac {3^{\frac {3}{4}} b^{\frac {4}{3}} c^{2} \EllipticF \left (\frac {a^{\frac {1}{3}} \left (1-\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}}{a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}}, i \sqrt {3}+2 i\right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right ) \sqrt {\frac {a^{\frac {2}{3}}+b^{\frac {2}{3}} c \,x^{2}-a^{\frac {1}{3}} b^{\frac {1}{3}} \sqrt {c \,x^{2}}}{\left (a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )^{2}}}\, \sqrt {2}}{8 a^{\frac {2}{3}} \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )}{\left (a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )^{2}}}}-\frac {3 \,3^{\frac {1}{4}} b^{\frac {4}{3}} c^{2} \EllipticE \left (\frac {a^{\frac {1}{3}} \left (1-\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}}{a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}}, i \sqrt {3}+2 i\right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}+b^{\frac {2}{3}} c \,x^{2}-a^{\frac {1}{3}} b^{\frac {1}{3}} \sqrt {c \,x^{2}}}{\left (a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )^{2}}}}{16 a^{\frac {2}{3}} \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )}{\left (a^{\frac {1}{3}} \left (1+\sqrt {3}\right )+b^{\frac {1}{3}} \sqrt {c \,x^{2}}\right )^{2}}}} \]
command
integrate((a+b*(c*x^2)^(3/2))^(1/2)/x^5,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, \sqrt {c x^{2}} \sqrt {\frac {\sqrt {c x^{2}} b c}{x}} b c x^{3} {\rm weierstrassZeta}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, x\right )\right ) + {\left (3 \, \sqrt {c x^{2}} b c x^{2} + 2 \, a\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{8 \, a x^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{x^{5}}, x\right ) \]