\[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (5-4 x \right ) \left (1+2 x \right )^{\frac {3}{2}}}{62 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {3 \left (11+78 x \right ) \sqrt {1+2 x}}{1922 \left (5 x^{2}+3 x +2\right )}+\frac {3 \ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right ) \sqrt {-4675420+838550 \sqrt {35}}}{595820}-\frac {3 \ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right ) \sqrt {-4675420+838550 \sqrt {35}}}{595820}-\frac {3 \arctan \left (\frac {-10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right ) \sqrt {4675420+838550 \sqrt {35}}}{297910}+\frac {3 \arctan \left (\frac {10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right ) \sqrt {4675420+838550 \sqrt {35}}}{297910} \]
command
integrate((1+2*x)**(5/2)/(5*x**2+3*x+2)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________