\[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {8 \left (1+2 x \right )^{\frac {3}{2}}}{155}-\frac {\left (5-4 x \right ) \left (1+2 x \right )^{\frac {5}{2}}}{31 \left (5 x^{2}+3 x +2\right )}+\frac {604 \sqrt {1+2 x}}{775}+\frac {\arctan \left (\frac {-10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right ) \sqrt {-1761642580+300382250 \sqrt {35}}}{120125}-\frac {\arctan \left (\frac {10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right ) \sqrt {-1761642580+300382250 \sqrt {35}}}{120125}+\frac {\ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right ) \sqrt {1761642580+300382250 \sqrt {35}}}{240250}-\frac {\ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right ) \sqrt {1761642580+300382250 \sqrt {35}}}{240250} \]
command
integrate((1+2*x)**(7/2)/(5*x**2+3*x+2)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {17280 \left (2 x + 1\right )^{\frac {3}{2}}}{- 124000 x + 77500 \left (2 x + 1\right )^{2} + 46500} + \frac {21280 \left (2 x + 1\right )^{\frac {3}{2}}}{- 868000 x + 542500 \left (2 x + 1\right )^{2} + 325500} + \frac {16 \sqrt {2 x + 1}}{25} + \frac {6912 \sqrt {2 x + 1}}{- 124000 x + 77500 \left (2 x + 1\right )^{2} + 46500} + \frac {57456 \sqrt {2 x + 1}}{- 868000 x + 542500 \left (2 x + 1\right )^{2} + 325500} + 16 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )} - \frac {1632 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} - \frac {624 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{125} - \frac {336 \operatorname {RootSum} {\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log {\left (- \frac {11049511452672 t^{3}}{2205125} + \frac {307918256 t}{2205125} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} - \frac {288 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} - \frac {112 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )}}{25} - \frac {48 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {48 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {48 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {448 \operatorname {RootSum} {\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log {\left (- \frac {11049511452672 t^{3}}{2205125} + \frac {307918256 t}{2205125} + \sqrt {2 x + 1} \right )} \right )\right )}}{125} + \frac {2016 \operatorname {RootSum} {\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log {\left (- \frac {11049511452672 t^{3}}{2205125} + \frac {307918256 t}{2205125} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} + \frac {3904 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )}}{125} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________