\[ \int \frac {\sqrt {1-2 x} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {203179 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{656250}-\frac {38723 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3609375}-\frac {2 \left (2+3 x \right )^{\frac {7}{2}} \sqrt {1-2 x}}{5 \sqrt {3+5 x}}+\frac {183 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{4375}+\frac {48 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{175}-\frac {2486 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{21875} \]
command
integrate((2+3*x)^(7/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (33750 \, x^{3} + 63225 \, x^{2} + 25955 \, x + 32\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{21875 \, \sqrt {5 \, x + 3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, x\right ) \]