\[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {488149 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{127575}-\frac {29357 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{255150}-\frac {223 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{945}-\frac {31 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{945}+\frac {2 \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{45}-\frac {29357 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{17010} \]
command
integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{17010} \, {\left (94500 \, x^{3} + 156150 \, x^{2} + 65250 \, x - 26009\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]