\[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (2 c d x +b d \right )^{\frac {3}{2}}}{2 \left (-4 a c +b^{2}\right ) d \left (c \,x^{2}+b x +a \right )^{2}}+\frac {5 c \left (2 c d x +b d \right )^{\frac {3}{2}}}{2 \left (-4 a c +b^{2}\right )^{2} d \left (c \,x^{2}+b x +a \right )}+\frac {5 c^{2} \arctan \left (\frac {\sqrt {d \left (2 c x +b \right )}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}\right ) \sqrt {d}}{\left (-4 a c +b^{2}\right )^{\frac {9}{4}}}-\frac {5 c^{2} \arctanh \left (\frac {\sqrt {d \left (2 c x +b \right )}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}\right ) \sqrt {d}}{\left (-4 a c +b^{2}\right )^{\frac {9}{4}}} \]
command
integrate((2*c*d*x+b*d)**(1/2)/(c*x**2+b*x+a)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {2304 a c^{3} d^{7} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{8192 a^{4} c^{4} d^{8} - 8192 a^{3} b^{2} c^{3} d^{8} + 4096 a^{3} c^{3} d^{6} \left (b d + 2 c d x\right )^{2} + 3072 a^{2} b^{4} c^{2} d^{8} - 3072 a^{2} b^{2} c^{2} d^{6} \left (b d + 2 c d x\right )^{2} + 512 a^{2} c^{2} d^{4} \left (b d + 2 c d x\right )^{4} - 512 a b^{6} c d^{8} + 768 a b^{4} c d^{6} \left (b d + 2 c d x\right )^{2} - 256 a b^{2} c d^{4} \left (b d + 2 c d x\right )^{4} + 32 b^{8} d^{8} - 64 b^{6} d^{6} \left (b d + 2 c d x\right )^{2} + 32 b^{4} d^{4} \left (b d + 2 c d x\right )^{4}} - \frac {576 b^{2} c^{2} d^{7} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{8192 a^{4} c^{4} d^{8} - 8192 a^{3} b^{2} c^{3} d^{8} + 4096 a^{3} c^{3} d^{6} \left (b d + 2 c d x\right )^{2} + 3072 a^{2} b^{4} c^{2} d^{8} - 3072 a^{2} b^{2} c^{2} d^{6} \left (b d + 2 c d x\right )^{2} + 512 a^{2} c^{2} d^{4} \left (b d + 2 c d x\right )^{4} - 512 a b^{6} c d^{8} + 768 a b^{4} c d^{6} \left (b d + 2 c d x\right )^{2} - 256 a b^{2} c d^{4} \left (b d + 2 c d x\right )^{4} + 32 b^{8} d^{8} - 64 b^{6} d^{6} \left (b d + 2 c d x\right )^{2} + 32 b^{4} d^{4} \left (b d + 2 c d x\right )^{4}} + \frac {320 c^{2} d^{5} \left (b d + 2 c d x\right )^{\frac {7}{2}}}{8192 a^{4} c^{4} d^{8} - 8192 a^{3} b^{2} c^{3} d^{8} + 4096 a^{3} c^{3} d^{6} \left (b d + 2 c d x\right )^{2} + 3072 a^{2} b^{4} c^{2} d^{8} - 3072 a^{2} b^{2} c^{2} d^{6} \left (b d + 2 c d x\right )^{2} + 512 a^{2} c^{2} d^{4} \left (b d + 2 c d x\right )^{4} - 512 a b^{6} c d^{8} + 768 a b^{4} c d^{6} \left (b d + 2 c d x\right )^{2} - 256 a b^{2} c d^{4} \left (b d + 2 c d x\right )^{4} + 32 b^{8} d^{8} - 64 b^{6} d^{6} \left (b d + 2 c d x\right )^{2} + 32 b^{4} d^{4} \left (b d + 2 c d x\right )^{4}} + 64 c^{2} d^{5} \operatorname {RootSum} {\left (t^{4} \cdot \left (70368744177664 a^{9} c^{9} d^{18} - 158329674399744 a^{8} b^{2} c^{8} d^{18} + 158329674399744 a^{7} b^{4} c^{7} d^{18} - 92358976733184 a^{6} b^{6} c^{6} d^{18} + 34634616274944 a^{5} b^{8} c^{5} d^{18} - 8658654068736 a^{4} b^{10} c^{4} d^{18} + 1443109011456 a^{3} b^{12} c^{3} d^{18} - 154618822656 a^{2} b^{14} c^{2} d^{18} + 9663676416 a b^{16} c d^{18} - 268435456 b^{18} d^{18}\right ) + 625, \left ( t \mapsto t \log {\left (\frac {34359738368 t^{3} a^{7} c^{7} d^{14}}{125} - \frac {60129542144 t^{3} a^{6} b^{2} c^{6} d^{14}}{125} + \frac {45097156608 t^{3} a^{5} b^{4} c^{5} d^{14}}{125} - \frac {3758096384 t^{3} a^{4} b^{6} c^{4} d^{14}}{25} + \frac {939524096 t^{3} a^{3} b^{8} c^{3} d^{14}}{25} - \frac {704643072 t^{3} a^{2} b^{10} c^{2} d^{14}}{125} + \frac {58720256 t^{3} a b^{12} c d^{14}}{125} - \frac {2097152 t^{3} b^{14} d^{14}}{125} + \sqrt {b d + 2 c d x} \right )} \right )\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________