\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^5(e+f x) \, dx \]
Optimal antiderivative \[ -\frac {\left (8 a^{2}+40 a b +35 b^{2}\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{24 \left (a +b \right )^{2} f}-\frac {\left (8 a +9 b \right ) \left (\sec ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{8 \left (a +b \right )^{2} f}+\frac {\left (\sec ^{4}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{4 \left (a +b \right ) f}+\frac {\left (8 a^{2}+40 a b +35 b^{2}\right ) \arctanh \! \left (\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sqrt {a +b}}\right )}{8 f \sqrt {a +b}}-\frac {\left (8 a^{2}+40 a b +35 b^{2}\right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{8 \left (a +b \right ) f} \]
command
integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)^5,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
\[ \text {output too large to display} \]