\[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {5 b \,\pi ^{\frac {5}{2}} x^{2}}{256 c}-\frac {59 b c \,\pi ^{\frac {5}{2}} x^{4}}{768}-\frac {17 b \,c^{3} \pi ^{\frac {5}{2}} x^{6}}{288}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{8}}{64}+\frac {5 \pi \,x^{3} \left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{48}+\frac {x^{3} \left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{8}-\frac {5 \pi ^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{256 b \,c^{3}}+\frac {5 \pi ^{\frac {5}{2}} x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{128 c^{2}}+\frac {5 \pi ^{2} x^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} \pi \,x^{2}+\pi }}{64} \]
command
int(x^2*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x)),x)
Maple 2022.1 output
\[\int x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
Maple 2021.1 output
\[ \frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{8 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{48 c^{2}}-\frac {5 a \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{192 c^{2}}-\frac {5 a \,\pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{128 c^{2}}-\frac {5 a \,\pi ^{3} \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{128 c^{2} \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {5}{2}} c^{4} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{7}}{8}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{8}}{64}+\frac {17 b \,\pi ^{\frac {5}{2}} c^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5}}{48}-\frac {17 b \,c^{3} \pi ^{\frac {5}{2}} x^{6}}{288}+\frac {59 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{192}-\frac {59 b c \,\pi ^{\frac {5}{2}} x^{4}}{768}+\frac {5 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{128 c^{2}}-\frac {5 b \,\pi ^{\frac {5}{2}} x^{2}}{256 c}-\frac {5 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right )^{2}}{256 c^{3}}+\frac {b \,\pi ^{\frac {5}{2}}}{72 c^{3}} \]