\[ \int \frac {1}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \cos \left (e x +d \right )-a \sin \left (e x +d \right )\right ) \left (b +a \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right )}{\left (a^{2}-b^{2}+c^{2}\right ) e \cos \left (e x +d \right )^{\frac {3}{2}} \left (a +b \sec \left (e x +d \right )+c \tan \left (e x +d \right )\right )^{\frac {3}{2}}}-\frac {2 \sqrt {\frac {\cos \left (d +e x -\arctan \left (a , c\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (a , c\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {a^{2}+c^{2}}}{b +\sqrt {a^{2}+c^{2}}}}\right ) \left (b +a \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right )^{2}}{\cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (a , c\right )}{2}\right ) \left (a^{2}-b^{2}+c^{2}\right ) e \cos \left (e x +d \right )^{\frac {3}{2}} \sqrt {\frac {b +a \cos \left (e x +d \right )+c \sin \left (e x +d \right )}{b +\sqrt {a^{2}+c^{2}}}}\, \left (a +b \sec \left (e x +d \right )+c \tan \left (e x +d \right )\right )^{\frac {3}{2}}} \]
command
integrate(1/cos(e*x+d)^(3/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a} \sqrt {\cos \left (e x + d\right )}}{b^{2} \cos \left (e x + d\right )^{2} \sec \left (e x + d\right )^{2} + c^{2} \cos \left (e x + d\right )^{2} \tan \left (e x + d\right )^{2} + 2 \, a b \cos \left (e x + d\right )^{2} \sec \left (e x + d\right ) + a^{2} \cos \left (e x + d\right )^{2} + 2 \, {\left (b c \cos \left (e x + d\right )^{2} \sec \left (e x + d\right ) + a c \cos \left (e x + d\right )^{2}\right )} \tan \left (e x + d\right )}, x\right ) \]