\[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {b \,\pi ^{\frac {3}{2}} x^{2}}{32 c}-\frac {7 b c \,\pi ^{\frac {3}{2}} x^{4}}{96}-\frac {b \,c^{3} \pi ^{\frac {3}{2}} x^{6}}{36}+\frac {x^{3} \left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{6}-\frac {\pi ^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{32 b \,c^{3}}+\frac {\pi ^{\frac {3}{2}} x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{16 c^{2}}+\frac {\pi \,x^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} \pi \,x^{2}+\pi }}{8} \]
command
int(x^2*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x)),x)
Maple 2022.1 output
\[\int x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
Maple 2021.1 output
\[ \frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{6 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{16 c^{2}}-\frac {a \,\pi ^{2} \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{16 c^{2} \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} c^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5}}{6}-\frac {b \,c^{3} \pi ^{\frac {3}{2}} x^{6}}{36}+\frac {7 b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{24}-\frac {7 b c \,\pi ^{\frac {3}{2}} x^{4}}{96}+\frac {b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{16 c^{2}}-\frac {b \,\pi ^{\frac {3}{2}} x^{2}}{32 c}-\frac {b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right )^{2}}{32 c^{3}}+\frac {b \,\pi ^{\frac {3}{2}}}{72 c^{3}} \]