\[ \int \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {25 b c \,\pi ^{\frac {5}{2}} x^{2}}{96}-\frac {5 b \,c^{3} \pi ^{\frac {5}{2}} x^{4}}{96}-\frac {b \,\pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{3}}{36 c}+\frac {5 \pi x \left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{24}+\frac {x \left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{6}+\frac {5 \pi ^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{32 b c}+\frac {5 \pi ^{2} x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} \pi \,x^{2}+\pi }}{16} \]
command
int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x)),x)
Maple 2022.1 output
\[\int \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
Maple 2021.1 output
\[ \frac {x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} a}{6}+\frac {5 a \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{24}+\frac {5 a \,\pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{16}+\frac {5 a \,\pi ^{3} \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{16 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {5}{2}} c^{4} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5}}{6}-\frac {b \,\pi ^{\frac {5}{2}} c^{5} x^{6}}{36}+\frac {13 b \,\pi ^{\frac {5}{2}} c^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{24}-\frac {13 b \,c^{3} \pi ^{\frac {5}{2}} x^{4}}{96}+\frac {11 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{16}-\frac {11 b c \,\pi ^{\frac {5}{2}} x^{2}}{32}+\frac {5 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right )^{2}}{32 c}-\frac {17 b \,\pi ^{\frac {5}{2}}}{72 c} \]