\[ \int \left (b \tan ^3(c+d x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 b^{2} \cot \left (d x +c \right ) \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}}{d}+\frac {b^{2} \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \sqrt {2}}{2 d \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {b^{2} \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \sqrt {2}}{2 d \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {b^{2} \ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \sqrt {2}}{4 d \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {b^{2} \ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \sqrt {2}}{4 d \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 b^{2} \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \tan \left (d x +c \right )}{5 d}-\frac {2 b^{2} \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \left (\tan ^{3}\left (d x +c \right )\right )}{9 d}+\frac {2 b^{2} \sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \left (\tan ^{5}\left (d x +c \right )\right )}{13 d} \]
command
integrate((b*tan(d*x+c)^3)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{2340} \, {\left (\frac {1170 \, \sqrt {2} b \sqrt {{\left | b \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{d} + \frac {1170 \, \sqrt {2} b \sqrt {{\left | b \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{d} + \frac {585 \, \sqrt {2} b \sqrt {{\left | b \right |}} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{d} - \frac {585 \, \sqrt {2} b \sqrt {{\left | b \right |}} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{d} + \frac {8 \, {\left (45 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{6} - 65 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{4} + 117 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{2} - 585 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12}\right )}}{b^{65} d^{13}}\right )} b \mathrm {sgn}\left (\tan \left (d x + c\right )\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \left (b \tan \left (d x + c\right )^{3}\right )^{\frac {5}{2}}\,{d x} \]________________________________________________________________________________________