\[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c+d \tan (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {4 \,\mathrm {I} a^{\frac {5}{2}} \arctanh \! \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {c -\mathrm {I} d}\, \sqrt {a +\mathrm {I} a \tan \left (f x +e \right )}}\right ) \sqrt {2}}{\left (c -\mathrm {I} d \right )^{\frac {5}{2}} f}+\frac {4 \,\mathrm {I} a^{2} \sqrt {a +\mathrm {I} a \tan \! \left (f x +e \right )}}{\left (c -\mathrm {I} d \right )^{2} f \sqrt {c +d \tan \! \left (f x +e \right )}}-\frac {2 a \left (a +\mathrm {I} a \tan \! \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (\mathrm {I} c +d \right ) f \left (c +d \tan \! \left (f x +e \right )\right )^{\frac {3}{2}}} \]
command
integrate((a+I*a*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Timed out} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {{\left (2 \, {\left (d \tan \left (f x + e\right ) + c\right )} a^{2} - 2 \, a^{2} c - 2 i \, a^{2} d\right )} \sqrt {2 \, a d^{2} + 2 \, \sqrt {{\left (d \tan \left (f x + e\right ) + c\right )}^{2} - 2 \, {\left (d \tan \left (f x + e\right ) + c\right )} c + c^{2} + d^{2}} a d} {\left (\frac {i \, {\left (d \tan \left (f x + e\right ) + c\right )} a d - i \, a c d}{a d^{2} + \sqrt {{\left (d \tan \left (f x + e\right ) + c\right )}^{2} a^{2} d^{2} - 2 \, {\left (d \tan \left (f x + e\right ) + c\right )} a^{2} c d^{2} + a^{2} c^{2} d^{2} + a^{2} d^{4}}} + 1\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{4 \, {\left ({\left (d \tan \left (f x + e\right ) + c\right )}^{3} d^{2} - {\left (d \tan \left (f x + e\right ) + c\right )}^{2} c d^{2} + i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} d^{3}\right )}} \]