\[ \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}}}{11 c \left (c x \right )^{\frac {11}{2}}}+\frac {2 b \left (-b \,x^{2}+a \right )^{\frac {1}{4}}}{77 a \,c^{3} \left (c x \right )^{\frac {7}{2}}}+\frac {4 b^{2} \left (-b \,x^{2}+a \right )^{\frac {1}{4}}}{77 a^{2} c^{5} \left (c x \right )^{\frac {3}{2}}}+\frac {8 b^{\frac {7}{2}} \left (1-\frac {a}{b \,x^{2}}\right )^{\frac {3}{4}} \left (c x \right )^{\frac {3}{2}} \sqrt {\frac {\sqrt {1-\frac {a}{b \,x^{2}}}}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {\mathrm {arccsc}\left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{2}\right ), \sqrt {2}\right )}{77 \cos \left (\frac {\mathrm {arccsc}\left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{2}\right ) a^{\frac {5}{2}} c^{8} \left (-b \,x^{2}+a \right )^{\frac {3}{4}}} \]
command
integrate((-b*x**2+a)**(1/4)/(c*x)**(13/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {i \sqrt [4]{b} e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a}{b x^{2}}} \right )}}{5 c^{\frac {13}{2}} x^{5}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________