\[ \int x \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx \]
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
integrate(x*(pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\pi ^{\frac {5}{2}} a c^{4} x^{6} \sqrt {c^{2} x^{2} + 1}}{7} + \frac {3 \pi ^{\frac {5}{2}} a c^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{7} + \frac {3 \pi ^{\frac {5}{2}} a x^{2} \sqrt {c^{2} x^{2} + 1}}{7} + \frac {\pi ^{\frac {5}{2}} a \sqrt {c^{2} x^{2} + 1}}{7 c^{2}} - \frac {\pi ^{\frac {5}{2}} b c^{5} x^{7}}{49} + \frac {\pi ^{\frac {5}{2}} b c^{4} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {3 \pi ^{\frac {5}{2}} b c^{3} x^{5}}{35} + \frac {3 \pi ^{\frac {5}{2}} b c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {\pi ^{\frac {5}{2}} b c x^{3}}{7} + \frac {3 \pi ^{\frac {5}{2}} b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {\pi ^{\frac {5}{2}} b x}{7 c} + \frac {\pi ^{\frac {5}{2}} b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7 c^{2}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {5}{2}} a x^{2}}{2} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________