\[ \int \left (b \tan ^3(c+d x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {2 b \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}}{3 d}+\frac {b \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \sqrt {2}}{2 d \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {b \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \sqrt {2}}{2 d \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {b \ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \sqrt {2}}{4 d \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {b \ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \sqrt {2}}{4 d \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 b \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \left (\tan ^{2}\left (d x +c \right )\right )}{7 d} \]
command
integrate((b*tan(d*x+c)^3)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{84} \, b {\left (\frac {42 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b d} + \frac {42 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b d} - \frac {21 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b d} + \frac {21 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b d} + \frac {8 \, {\left (3 \, \sqrt {b \tan \left (d x + c\right )} b^{21} d^{6} \tan \left (d x + c\right )^{3} - 7 \, \sqrt {b \tan \left (d x + c\right )} b^{21} d^{6} \tan \left (d x + c\right )\right )}}{b^{21} d^{7}}\right )} \mathrm {sgn}\left (\tan \left (d x + c\right )\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \left (b \tan \left (d x + c\right )^{3}\right )^{\frac {3}{2}}\,{d x} \]________________________________________________________________________________________