\[ \int \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx \]
Optimal antiderivative \[ -\frac {2911577 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1771875}-\frac {175111 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3543750}-\frac {23 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{1575}+\frac {2 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{45}-\frac {1244 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{13125}-\frac {175111 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{236250} \]
command
integrate((2+3*x)^(5/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{236250} \, {\left (472500 \, x^{3} + 861750 \, x^{2} + 410490 \, x - 136987\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]