\[ \int x^3 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {\left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{3 c^{4} \pi }+\frac {\left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{5 c^{4} \pi ^{2}}+\frac {2 b x \sqrt {\pi }}{15 c^{3}}-\frac {b \,x^{3} \sqrt {\pi }}{45 c}-\frac {b c \,x^{5} \sqrt {\pi }}{25} \]
command
int(x^3*(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2),x)
Maple 2022.1 output
\[\int x^{3} \left (a +b \arcsinh \left (c x \right )\right ) \sqrt {\pi \,c^{2} x^{2}+\pi }\, dx\]
Maple 2021.1 output
\[ a \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{5 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{15 \pi \,c^{4}}\right )+\frac {b \sqrt {\pi }\, \left (45 \arcsinh \left (c x \right ) c^{6} x^{6}+60 \arcsinh \left (c x \right ) c^{4} x^{4}-9 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}-15 \arcsinh \left (c x \right ) c^{2} x^{2}-5 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-30 \arcsinh \left (c x \right )+30 c x \sqrt {c^{2} x^{2}+1}\right )}{225 c^{4} \sqrt {c^{2} x^{2}+1}} \]