\[ \int x \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {b \,\pi ^{\frac {5}{2}} x}{7 c}-\frac {b c \,\pi ^{\frac {5}{2}} x^{3}}{7}-\frac {3 b \,c^{3} \pi ^{\frac {5}{2}} x^{5}}{35}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{7}}{49}+\frac {\left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {7}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{7 c^{2} \pi } \]
command
int(x*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x)),x)
Maple 2022.1 output
\[\int x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
Maple 2021.1 output
\[ \frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{7 \pi \,c^{2}}+\frac {b \,\pi ^{\frac {5}{2}} \left (35 \arcsinh \left (c x \right ) c^{8} x^{8}+140 \arcsinh \left (c x \right ) c^{6} x^{6}-5 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+210 \arcsinh \left (c x \right ) c^{4} x^{4}-21 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+140 \arcsinh \left (c x \right ) c^{2} x^{2}-35 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+35 \arcsinh \left (c x \right )-35 c x \sqrt {c^{2} x^{2}+1}\right )}{245 c^{2} \sqrt {c^{2} x^{2}+1}} \]