\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {4418 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1215}+\frac {988 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1215}-\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \sqrt {3+5 x}}{9 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {10 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}+\frac {196 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{81} \]
command
integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (36 \, x^{2} + 1077 \, x + 653\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{81 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}, x\right ) \]