\[ \int \frac {\sec ^5(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {4 \sqrt {\frac {1}{2}+\frac {\sin \left (2 f x +2 e \right )}{2}}\, \EllipticF \left (\cos \left (e +\frac {\pi }{4}+f x \right ), \sqrt {2}\right ) \sec \left (f x +e \right ) \left (\sqrt {\sin }\left (2 f x +2 e \right )\right )}{7 \sin \left (e +\frac {\pi }{4}+f x \right ) f \sqrt {d \tan \left (f x +e \right )}}+\frac {4 \sec \left (f x +e \right ) \sqrt {d \tan \left (f x +e \right )}}{7 d f}+\frac {2 \left (\sec ^{3}\left (f x +e \right )\right ) \sqrt {d \tan \left (f x +e \right )}}{7 d f} \]
command
integrate(sec(f*x+e)^5/(d*tan(f*x+e))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, \sqrt {i \, d} \cos \left (f x + e\right )^{3} {\rm ellipticF}\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ), -1\right ) + 2 \, \sqrt {-i \, d} \cos \left (f x + e\right )^{3} {\rm ellipticF}\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ), -1\right ) - {\left (2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}\right )}}{7 \, d f \cos \left (f x + e\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d \tan \left (f x + e\right )} \sec \left (f x + e\right )^{5}}{d \tan \left (f x + e\right )}, x\right ) \]