Question Number 77817 by Dah Solu Tion last updated on 10/Jan/20
$$\int\frac{{dx}}{{x}^{\mathrm{3}} \left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}\right)^{\mathrm{4}} } \\ $$$$ \\ $$
Commented by mathmax by abdo last updated on 10/Jan/20
$${we}\:{decompose}\:{F}\left({x}\right)=\frac{\mathrm{1}}{{x}^{\mathrm{3}} \left({x}^{\mathrm{2}} \:+\mathrm{2}{x}+\mathrm{5}\right)^{\mathrm{4}} }\:{inside}\:{C}\left({x}\right) \\ $$$${x}^{\mathrm{2}\:} +\mathrm{2}{x}\:+\mathrm{5}\:=\mathrm{0}\:\rightarrow\Delta^{'} =\mathrm{1}−\mathrm{5}=−\mathrm{4}\:\Rightarrow{x}_{\mathrm{1}} =−\mathrm{1}+\mathrm{2}{i}\:{and}\:{x}_{\mathrm{2}} =−\mathrm{1}−\mathrm{2}{i} \\ $$$${x}_{\mathrm{1}} =\sqrt{\mathrm{5}}{e}^{{iarctan}\left(−\mathrm{2}\right)} \:=\sqrt{\mathrm{5}}{e}^{−{iarctan}\left(\mathrm{2}\right)} \:{and}\:{x}_{\mathrm{2}} =\sqrt{\mathrm{5}}{e}^{{i}\:{arctan}\left(\mathrm{2}\right)} \:\Rightarrow \\ $$$${F}\left({x}\right)=\frac{\mathrm{1}}{{x}^{\mathrm{3}} \left({x}−\sqrt{\mathrm{5}}{e}^{{iarctan}\left(\mathrm{2}\right)} \right)^{\mathrm{4}} \left({x}−\sqrt{\mathrm{5}}{e}^{−{iarctan}\left(\mathrm{2}\right)} \right)^{\mathrm{4}} } \\ $$$$=\frac{{a}}{{x}}\:+\frac{{b}}{{x}^{\mathrm{2}} }\:+\frac{{c}}{{x}^{\mathrm{3}} }\:+\sum_{{i}=\mathrm{1}} ^{\mathrm{4}} \:\frac{{a}_{{i}} }{\left({x}−\sqrt{\mathrm{5}}{e}^{{iarctan}\left(\mathrm{2}\right)} \right)^{{i}} }\:+\sum_{{i}=\mathrm{1}} ^{\mathrm{4}} \:\frac{{b}_{{i}} }{\left({x}−\sqrt{\mathrm{5}}{e}^{−{iarctan}\left(\mathrm{2}\right)} \right)^{{i}} } \\ $$$$\Rightarrow\int\:{F}\left({x}\right){dx}\:={aln}\mid{x}\mid−\frac{{b}}{{x}}\:−\frac{{c}}{\mathrm{2}{x}^{\mathrm{2}} }\:+\sum_{{i}=\mathrm{1}} ^{\mathrm{4}} {a}_{{i}} \:\int\:\:\frac{{dx}}{\left({x}−\sqrt{\mathrm{5}}{e}^{{iarctan}\left(\mathrm{2}\right)} \right)^{{i}} } \\ $$$$+\sum_{{i}=\mathrm{1}} ^{\mathrm{4}} \:{b}_{{i}} \int\:\:\:\:\:\frac{{dx}}{\left({x}−\sqrt{\mathrm{5}}{e}^{−{i}\:{arctan}\left(\mathrm{2}\right)} \right)^{{i}} }\:+{C}\: \\ $$$${rest}\:{to}\:{find}\:{a}_{{i}} \:{and}\:{b}_{{i}} ….{be}\:{continued}\:… \\ $$
Commented by MJS last updated on 11/Jan/20
$$\mathrm{look}\:\mathrm{up}\:\mathrm{Ostrogradski}'\mathrm{s}\:\mathrm{method}\:\mathrm{on}\:\mathrm{the}\:\mathrm{web}; \\ $$$$\mathrm{I}\:\mathrm{think}\:\mathrm{it}'\mathrm{s}\:\mathrm{the}\:\mathrm{best}\:\mathrm{in}\:\mathrm{cases}\:\mathrm{like}\:\mathrm{this}\:\mathrm{one}.\:\mathrm{I}\:\mathrm{have} \\ $$$$\mathrm{no}\:\mathrm{time}\:\mathrm{now},\:\mathrm{will}\:\mathrm{post}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{later} \\ $$
Commented by Dah Solu Tion last updated on 11/Jan/20
$${Thanks}\:{man} \\ $$$${i}\:{will}\:{try}\:{d}\:{rest} \\ $$