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