\[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^3} \, dx \]
Optimal antiderivative \[ \frac {\left (a -b \right ) \left (a^{2}+4 a b +b^{2}\right ) \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{2 \left (a^{2}+b^{2}\right )^{3} d}+\frac {\left (a -b \right ) \left (a^{2}+4 a b +b^{2}\right ) \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{2 \left (a^{2}+b^{2}\right )^{3} d}+\frac {\left (a +b \right ) \left (a^{2}-4 a b +b^{2}\right ) \ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{4 \left (a^{2}+b^{2}\right )^{3} d}-\frac {\left (a +b \right ) \left (a^{2}-4 a b +b^{2}\right ) \ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{4 \left (a^{2}+b^{2}\right )^{3} d}-\frac {\left (15 a^{4}-18 a^{2} b^{2}-b^{4}\right ) \arctan \left (\frac {\sqrt {b}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a}}\right ) \sqrt {b}}{4 a^{\frac {3}{2}} \left (a^{2}+b^{2}\right )^{3} d}-\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{2 \left (a^{2}+b^{2}\right ) d \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {b \left (7 a^{2}-b^{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{4 a \left (a^{2}+b^{2}\right )^{2} d \left (a +b \tan \left (d x +c \right )\right )} \]
command
integrate(tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^3,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 \[ \text {Timed out} \]