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