\[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {829177897 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1052493750}-\frac {12996374 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{526246875}+\frac {326 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}} \sqrt {2+3 x}}{7425}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}} \sqrt {2+3 x}}{55}-\frac {78797 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{3898125}+\frac {30362 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{779625}-\frac {12996374 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{35083125} \]
command
integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)*(2+3*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{35083125} \, {\left (127575000 \, x^{4} - 51502500 \, x^{3} - 95024250 \, x^{2} + 48272535 \, x + 22517617\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]