\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^5} \, dx \]
Optimal antiderivative \[ \frac {2 a^{\frac {5}{2}} \arctan \left (\frac {\sqrt {a}\, \tan \left (f x +e \right )}{\sqrt {a +a \sec \left (f x +e \right )}}\right )}{c^{5} f}-\frac {2 a \left (\cot ^{3}\left (f x +e \right )\right ) \left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{3 c^{5} f}+\frac {2 \left (\cot ^{5}\left (f x +e \right )\right ) \left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}}}{5 c^{5} f}+\frac {8 \left (\cot ^{9}\left (f x +e \right )\right ) \left (a +a \sec \left (f x +e \right )\right )^{\frac {9}{2}}}{9 a^{2} c^{5} f}+\frac {2 a^{2} \cot \left (f x +e \right ) \sqrt {a +a \sec \left (f x +e \right )}}{c^{5} f} \]
command
integrate((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {45 \, \sqrt {-a} a^{3} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{c^{5} {\left | a \right |}} + \frac {4 \, {\left (45 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{16} \sqrt {-a} a^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 270 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{14} \sqrt {-a} a^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 900 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{12} \sqrt {-a} a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 1575 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{10} \sqrt {-a} a^{6} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1953 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{8} \sqrt {-a} a^{7} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 1452 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{6} \sqrt {-a} a^{8} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 738 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{4} \sqrt {-a} a^{9} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 207 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} \sqrt {-a} a^{10} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 28 \, \sqrt {2} \sqrt {-a} a^{11} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - a\right )}^{9} c^{5}}}{45 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________