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