\[ \int \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx \]
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
integrate((pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\pi ^{\frac {5}{2}} a c^{4} x^{5} \sqrt {c^{2} x^{2} + 1}}{6} + \frac {13 \pi ^{\frac {5}{2}} a c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{24} + \frac {11 \pi ^{\frac {5}{2}} a x \sqrt {c^{2} x^{2} + 1}}{16} + \frac {5 \pi ^{\frac {5}{2}} a \operatorname {asinh}{\left (c x \right )}}{16 c} - \frac {\pi ^{\frac {5}{2}} b c^{5} x^{6}}{36} + \frac {\pi ^{\frac {5}{2}} b c^{4} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{6} - \frac {13 \pi ^{\frac {5}{2}} b c^{3} x^{4}}{96} + \frac {13 \pi ^{\frac {5}{2}} b c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{24} - \frac {11 \pi ^{\frac {5}{2}} b c x^{2}}{32} + \frac {11 \pi ^{\frac {5}{2}} b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16} + \frac {5 \pi ^{\frac {5}{2}} b \operatorname {asinh}^{2}{\left (c x \right )}}{32 c} & \text {for}\: c \neq 0 \\\pi ^{\frac {5}{2}} a x & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________