\[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx \]
Optimal antiderivative \[ \frac {1576 \sqrt {1+\frac {1}{a x}}\, \sqrt {-a c x +c}}{315 a^{4} \sqrt {1-\frac {1}{a x}}}+\frac {472 x \sqrt {1+\frac {1}{a x}}\, \sqrt {-a c x +c}}{315 a^{3} \sqrt {1-\frac {1}{a x}}}+\frac {92 x^{2} \sqrt {1+\frac {1}{a x}}\, \sqrt {-a c x +c}}{105 a^{2} \sqrt {1-\frac {1}{a x}}}+\frac {38 x^{3} \sqrt {1+\frac {1}{a x}}\, \sqrt {-a c x +c}}{63 a \sqrt {1-\frac {1}{a x}}}+\frac {2 x^{4} \sqrt {1+\frac {1}{a x}}\, \sqrt {-a c x +c}}{9 \sqrt {1-\frac {1}{a x}}}-\frac {4 \arctanh \! \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{x}}}{\sqrt {a}\, \sqrt {1+\frac {1}{a x}}}\right ) \sqrt {2}\, \sqrt {\frac {1}{x}}\, \sqrt {-a c x +c}}{a^{\frac {9}{2}} \sqrt {1-\frac {1}{a x}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)*x^3*(-a*c*x+c)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {\frac {1260 i \, \sqrt {2} \sqrt {-c} \arctan \left (-i\right ) - 2584 \, \sqrt {2} \sqrt {-c}}{a^{3} \mathrm {sgn}\left (c\right )} + \frac {2 \, {\left (630 \, \sqrt {2} c^{\frac {9}{2}} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right ) - 35 \, {\left (a c x + c\right )}^{4} \sqrt {-a c x - c} + 45 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} c - 63 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c^{2} + 105 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{3} - 630 \, \sqrt {-a c x - c} c^{4}\right )}}{a^{3} c^{4} \mathrm {sgn}\left (-a c x - c\right )}}{315 \, a} \]