\[ \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx \]
Optimal antiderivative \[ -\frac {8 b x}{15 c^{5} \sqrt {\pi }}+\frac {4 b \,x^{3}}{45 c^{3} \sqrt {\pi }}-\frac {b \,x^{5}}{25 c \sqrt {\pi }}+\frac {8 \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} \pi \,x^{2}+\pi }}{15 c^{6} \pi }-\frac {4 x^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} \pi \,x^{2}+\pi }}{15 c^{4} \pi }+\frac {x^{4} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} \pi \,x^{2}+\pi }}{5 c^{2} \pi } \]
command
int(x^5*(a+b*arcsinh(c*x))/(Pi*c^2*x^2+Pi)^(1/2),x)
Maple 2022.1 output
\[\int \frac {x^{5} \left (a +b \arcsinh \left (c x \right )\right )}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\, dx\]
Maple 2021.1 output
\[ a \left (\frac {x^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}{5 \pi \,c^{2}}-\frac {4 \left (\frac {x^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{2}}-\frac {2 \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{4}}\right )}{5 c^{2}}\right )+\frac {b \left (45 \arcsinh \left (c x \right ) c^{6} x^{6}-15 \arcsinh \left (c x \right ) c^{4} x^{4}-9 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+60 \arcsinh \left (c x \right ) c^{2} x^{2}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+120 \arcsinh \left (c x \right )-120 c x \sqrt {c^{2} x^{2}+1}\right )}{225 c^{6} \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}} \]