\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx \]
Optimal antiderivative \[ -\frac {231061879 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{584718750}-\frac {3963068 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{292359375}+\frac {62 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {5}{2}} \sqrt {3+5 x}}{1485}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (2+3 x \right )^{\frac {5}{2}} \sqrt {3+5 x}}{55}+\frac {181333 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{3898125}+\frac {4258 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{155925}-\frac {2865161 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{19490625} \]
command
integrate((1-2*x)^(5/2)*(2+3*x)^(5/2)/(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{19490625} \, {\left (25515000 \, x^{4} - 6142500 \, x^{3} - 23717250 \, x^{2} + 9526995 \, x + 7167169\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{\sqrt {5 \, x + 3}}, x\right ) \]