\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2 \, dx \]
Optimal antiderivative \[ \frac {5592499 \EllipticF \left (\frac {\sqrt {33}\, \sqrt {1+4 x}}{11}, \frac {\sqrt {3}}{3}\right ) \sqrt {66}\, \sqrt {5-2 x}}{23328 \sqrt {-5+2 x}}-\frac {17746949 \EllipticE \left (\frac {2 \sqrt {2-3 x}\, \sqrt {11}}{11}, \frac {i \sqrt {2}}{2}\right ) \sqrt {11}\, \sqrt {-5+2 x}}{29160 \sqrt {5-2 x}}-\frac {5256763 \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}{97200}-\frac {8141 \left (7+5 x \right ) \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}{2700}-\frac {61 \left (7+5 x \right )^{2} \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}{270}+\frac {2 \left (7+5 x \right )^{3} \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}{45} \]
command
integrate((7+5*x)^2*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{19440} \, {\left (108000 \, x^{3} + 343800 \, x^{2} + 34524 \, x - 1380515\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (25 \, x^{2} + 70 \, x + 49\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}, x\right ) \]