\[ \int \frac {(b d+2 c d x)^{5/2}}{a+b x+c x^2} \, dx \]
Optimal antiderivative \[ \frac {4 d \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+2 \left (-4 a c +b^{2}\right )^{\frac {3}{4}} d^{\frac {5}{2}} \arctan \! \left (\frac {\sqrt {d \left (2 c x +b \right )}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}\right )-2 \left (-4 a c +b^{2}\right )^{\frac {3}{4}} d^{\frac {5}{2}} \arctanh \! \left (\frac {\sqrt {d \left (2 c x +b \right )}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}\right ) \]
command
integrate((2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ 32 a b c d^{4} \operatorname {RootSum} {\left (t^{4} \left (16384 a^{3} c^{3} d^{6} - 12288 a^{2} b^{2} c^{2} d^{6} + 3072 a b^{4} c d^{6} - 256 b^{6} d^{6}\right ) + 1, \left ( t \mapsto t \log {\left (16 t a c d^{2} - 4 t b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} - 32 a b c d^{4} \operatorname {RootSum} {\left (t^{4} \left (16384 a^{3} c^{3} d^{6} - 12288 a^{2} b^{2} c^{2} d^{6} + 3072 a b^{4} c d^{6} - 256 b^{6} d^{6}\right ) + 1, \left ( t \mapsto t \log {\left (16 t a c d^{2} - 4 t b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} - 16 a c d^{3} \operatorname {RootSum} {\left (t^{4} \left (1024 a c d^{2} - 256 b^{2} d^{2}\right ) + 1, \left ( t \mapsto t \log {\left (256 t^{3} a c d^{2} - 64 t^{3} b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} - 8 b^{3} d^{4} \operatorname {RootSum} {\left (t^{4} \left (16384 a^{3} c^{3} d^{6} - 12288 a^{2} b^{2} c^{2} d^{6} + 3072 a b^{4} c d^{6} - 256 b^{6} d^{6}\right ) + 1, \left ( t \mapsto t \log {\left (16 t a c d^{2} - 4 t b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} + 8 b^{3} d^{4} \operatorname {RootSum} {\left (t^{4} \left (16384 a^{3} c^{3} d^{6} - 12288 a^{2} b^{2} c^{2} d^{6} + 3072 a b^{4} c d^{6} - 256 b^{6} d^{6}\right ) + 1, \left ( t \mapsto t \log {\left (16 t a c d^{2} - 4 t b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} + 8 b^{2} d^{3} \operatorname {RootSum} {\left (t^{4} \left (1024 a c d^{2} - 256 b^{2} d^{2}\right ) + 1, \left ( t \mapsto t \log {\left (256 t^{3} a c d^{2} - 64 t^{3} b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} - 8 b^{2} d^{3} \operatorname {RootSum} {\left (t^{4} \left (1024 a c d^{2} - 256 b^{2} d^{2}\right ) + 1, \left ( t \mapsto t \log {\left (256 t^{3} a c d^{2} - 64 t^{3} b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} + 4 b^{2} d^{3} \operatorname {RootSum} {\left (t^{4} \left (1024 a c d^{2} - 256 b^{2} d^{2}\right ) + 1, \left ( t \mapsto t \log {\left (256 t^{3} a c d^{2} - 64 t^{3} b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} + \frac {4 d \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} \]