\[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (i A +B \right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {-i b +a}}\right )}{\left (-i b +a \right )^{\frac {5}{2}} d}+\frac {\left (i A -B \right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {i b +a}}\right )}{\left (i b +a \right )^{\frac {5}{2}} d}+\frac {2 \left (8 A \,a^{4} b +17 A \,a^{2} b^{3}+3 A \,b^{5}-16 B \,a^{5}-30 B \,a^{3} b^{2}-8 B a \,b^{4}\right ) \sqrt {a +b \tan \left (d x +c \right )}}{3 b^{4} \left (a^{2}+b^{2}\right )^{2} d}-\frac {2 \left (4 A \,a^{3} b +10 A a \,b^{3}-8 a^{4} B -15 B \,a^{2} b^{2}-b^{4} B \right ) \sqrt {a +b \tan \left (d x +c \right )}\, \tan \left (d x +c \right )}{3 b^{3} \left (a^{2}+b^{2}\right )^{2} d}+\frac {2 a \left (A \,a^{2} b +3 A \,b^{3}-2 a^{3} B -4 B a \,b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{b^{2} \left (a^{2}+b^{2}\right )^{2} d \sqrt {a +b \tan \left (d x +c \right )}}+\frac {2 a \left (A b -B a \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 b \left (a^{2}+b^{2}\right ) d \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}} \]
command
integrate(tan(d*x+c)^4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]