\[ \int x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ \frac {\left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{3 c^{2} \pi }-\frac {b x \sqrt {\pi }}{3 c}-\frac {b c \,x^{3} \sqrt {\pi }}{9} \]
command
int(x*(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2),x)
Maple 2022.1 output
\[\int x \left (a +b \arcsinh \left (c x \right )\right ) \sqrt {\pi \,c^{2} x^{2}+\pi }\, dx\]
Maple 2021.1 output
\[ \frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,c^{2}}+\frac {b \sqrt {\pi }\, \left (3 \arcsinh \left (c x \right ) c^{4} x^{4}+6 \arcsinh \left (c x \right ) c^{2} x^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+3 \arcsinh \left (c x \right )-3 c x \sqrt {c^{2} x^{2}+1}\right )}{9 c^{2} \sqrt {c^{2} x^{2}+1}} \]