\[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {b^{2} \pi ^{\frac {3}{2}} x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{32}-\frac {9 b^{2} \pi ^{\frac {3}{2}} \arcsinh \! \left (c x \right )}{64 c}-\frac {3 b c \,\pi ^{\frac {3}{2}} x^{2} \left (a +b \arcsinh \! \left (c x \right )\right )}{8}-\frac {b \,\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )}{8 c}+\frac {x \left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{4}+\frac {\pi ^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )^{3}}{8 b c}+\frac {15 b^{2} \pi ^{\frac {3}{2}} x \sqrt {c^{2} x^{2}+1}}{64}+\frac {3 \pi x \left (a +b \arcsinh \! \left (c x \right )\right )^{2} \sqrt {c^{2} \pi \,x^{2}+\pi }}{8} \]
command
int((Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {a^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a^{2} \pi ^{2} \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}+\frac {b^{2} \pi ^{\frac {3}{2}} c^{2} \arcsinh \left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3}}{4}-\frac {b^{2} \pi ^{\frac {3}{2}} c^{3} \arcsinh \left (c x \right ) x^{4}}{8}+\frac {b^{2} \pi ^{\frac {3}{2}} c^{2} x^{3} \sqrt {c^{2} x^{2}+1}}{32}+\frac {5 b^{2} \pi ^{\frac {3}{2}} \arcsinh \left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x}{8}-\frac {5 b^{2} \pi ^{\frac {3}{2}} c \arcsinh \left (c x \right ) x^{2}}{8}+\frac {17 b^{2} \pi ^{\frac {3}{2}} x \sqrt {c^{2} x^{2}+1}}{64}+\frac {b^{2} \pi ^{\frac {3}{2}} \arcsinh \left (c x \right )^{3}}{8 c}-\frac {17 b^{2} \pi ^{\frac {3}{2}} \arcsinh \left (c x \right )}{64 c}+\frac {a b \,\pi ^{\frac {3}{2}} c^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{2}-\frac {a b \,\pi ^{\frac {3}{2}} c^{3} x^{4}}{8}+\frac {5 a b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{4}-\frac {5 a b \,\pi ^{\frac {3}{2}} c \,x^{2}}{8}+\frac {3 a b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right )^{2}}{8 c}-\frac {a b \,\pi ^{\frac {3}{2}}}{2 c} \]