\[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {8120161139 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{4104725625}-\frac {486785077 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{8209451250}+\frac {326 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {7}{2}} \sqrt {2+3 x}}{10725}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {7}{2}} \sqrt {2+3 x}}{65}-\frac {3872003 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{30405375}-\frac {121031 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{30405375}+\frac {2314 \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{111375}-\frac {486785077 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{547296750} \]
command
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)*(2+3*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{547296750} \, {\left (8419950000 \, x^{5} + 2577015000 \, x^{4} - 7942630500 \, x^{3} - 1730459250 \, x^{2} + 2923422930 \, x + 495379991\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]