\[ \int \frac {\sec ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \tan \left (f x +e \right )}{3 \left (a +b \right )^{2} f \sqrt {a +b +b \left (\tan ^{2}\left (f x +e \right )\right )}}+\frac {\tan \left (f x +e \right )}{3 \left (a +b \right ) f \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}} \]
command
integrate(sec(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (\frac {3 \, {\left (a^{6} b^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 2 \, a^{5} b^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + a^{4} b^{6} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}} - \frac {2 \, {\left (3 \, a^{6} b^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 2 \, a^{5} b^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - a^{4} b^{6} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )}}{a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \frac {3 \, {\left (a^{6} b^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 2 \, a^{5} b^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + a^{4} b^{6} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )}}{a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{3 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a + b\right )}^{\frac {3}{2}} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________