\[ \int \sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {\left (a -b \right )^{2} \left (a +5 b \right ) \arctanh \left (\frac {\cosh \left (f x +e \right ) \sqrt {b}}{\sqrt {a -b +b \left (\cosh ^{2}\left (f x +e \right )\right )}}\right )}{16 b^{\frac {3}{2}} f}-\frac {\left (a +5 b \right ) \cosh \left (f x +e \right ) \left (a -b +b \left (\cosh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{24 b f}+\frac {\cosh \left (f x +e \right ) \left (a -b +b \left (\cosh ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{6 b f}-\frac {\left (a -b \right ) \left (a +5 b \right ) \cosh \left (f x +e \right ) \sqrt {a -b +b \left (\cosh ^{2}\left (f x +e \right )\right )}}{16 b f} \]
command
integrate(sinh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: TypeError} \]_______________________________________________________