\[ \int x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {b \,\pi ^{\frac {3}{2}} x}{5 c}-\frac {2 b c \,\pi ^{\frac {3}{2}} x^{3}}{15}-\frac {b \,c^{3} \pi ^{\frac {3}{2}} x^{5}}{25}+\frac {\left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{5 c^{2} \pi } \]
command
int(x*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x)),x)
Maple 2022.1 output
\[\int x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
Maple 2021.1 output
\[ \frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{5 \pi \,c^{2}}+\frac {b \,\pi ^{\frac {3}{2}} \left (15 \arcsinh \left (c x \right ) c^{6} x^{6}+45 \arcsinh \left (c x \right ) c^{4} x^{4}-3 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+45 \arcsinh \left (c x \right ) c^{2} x^{2}-10 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+15 \arcsinh \left (c x \right )-15 c x \sqrt {c^{2} x^{2}+1}\right )}{75 c^{2} \sqrt {c^{2} x^{2}+1}} \]