Integrand size = 14, antiderivative size = 104 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=735 x+\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {x^2 (2206+1593 x)}{2 \left (2+3 x+x^2\right )}-1471 \log (1+x)+1472 \log (2+x) \]
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Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {752, 832, 787, 646, 31} \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=-\frac {(1593 x+2206) x^2}{2 \left (x^2+3 x+2\right )}+\frac {(3 x+4) x^8}{4 \left (x^2+3 x+2\right )^4}-\frac {(81 x+110) x^6}{12 \left (x^2+3 x+2\right )^3}+\frac {(135 x+184) x^4}{2 \left (x^2+3 x+2\right )^2}+735 x-1471 \log (x+1)+1472 \log (x+2) \]
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Rule 31
Rule 646
Rule 752
Rule 787
Rule 832
Rubi steps \begin{align*} \text {integral}& = \frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {1}{4} \int \frac {x^7 (32+3 x)}{\left (2+3 x+x^2\right )^4} \, dx \\ & = \frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}-\frac {1}{12} \int \frac {(-660-72 x) x^5}{\left (2+3 x+x^2\right )^3} \, dx \\ & = \frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {1}{24} \int \frac {x^3 (8832+1476 x)}{\left (2+3 x+x^2\right )^2} \, dx \\ & = \frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {x^2 (2206+1593 x)}{2 \left (2+3 x+x^2\right )}-\frac {1}{24} \int \frac {(-52944-17640 x) x}{2+3 x+x^2} \, dx \\ & = 735 x+\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {x^2 (2206+1593 x)}{2 \left (2+3 x+x^2\right )}-\frac {1}{24} \int \frac {35280-24 x}{2+3 x+x^2} \, dx \\ & = 735 x+\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {x^2 (2206+1593 x)}{2 \left (2+3 x+x^2\right )}-1471 \int \frac {1}{1+x} \, dx+1472 \int \frac {1}{2+x} \, dx \\ & = 735 x+\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {x^2 (2206+1593 x)}{2 \left (2+3 x+x^2\right )}-1471 \log (1+x)+1472 \log (2+x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=\frac {514+513 x}{4 \left (2+3 x+x^2\right )^4}+\frac {415+1998 x}{12 \left (2+3 x+x^2\right )^3}+\frac {3 (451+456 x)}{4 \left (2+3 x+x^2\right )^2}-\frac {2 (1114+729 x)}{2+3 x+x^2}-1471 \log (1+x)+1472 \log (2+x) \]
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Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.58
method | result | size |
norman | \(\frac {-229950 x^{3}-85880 x -67824 x^{5}-15350 x^{6}-1458 x^{7}-\frac {651951}{4} x^{4}-\frac {571502}{3} x^{2}-\frac {48820}{3}}{\left (x^{2}+3 x +2\right )^{4}}-1471 \ln \left (1+x \right )+1472 \ln \left (2+x \right )\) | \(60\) |
risch | \(\frac {-229950 x^{3}-85880 x -67824 x^{5}-15350 x^{6}-1458 x^{7}-\frac {651951}{4} x^{4}-\frac {571502}{3} x^{2}-\frac {48820}{3}}{\left (x^{2}+3 x +2\right )^{4}}-1471 \ln \left (1+x \right )+1472 \ln \left (2+x \right )\) | \(60\) |
default | \(-\frac {128}{\left (2+x \right )^{4}}-\frac {256}{3 \left (2+x \right )^{3}}-\frac {384}{\left (2+x \right )^{2}}-\frac {1024}{2+x}+1472 \ln \left (2+x \right )+\frac {1}{4 \left (1+x \right )^{4}}-\frac {14}{3 \left (1+x \right )^{3}}+\frac {48}{\left (1+x \right )^{2}}-\frac {434}{1+x}-1471 \ln \left (1+x \right )\) | \(70\) |
parallelrisch | \(-\frac {195280+184200 x^{6}+1030560 x +813888 x^{5}+282432 \ln \left (1+x \right )-282624 \ln \left (2+x \right )+1955853 x^{4}+2759400 x^{3}+2286008 x^{2}+17496 x^{7}-1695744 \ln \left (2+x \right ) x +1094424 \ln \left (1+x \right ) x^{6}-1095168 \ln \left (2+x \right ) x^{6}+3177360 \ln \left (1+x \right ) x^{5}+17652 \ln \left (1+x \right ) x^{8}-17664 \ln \left (2+x \right ) x^{8}+211824 \ln \left (1+x \right ) x^{7}-211968 \ln \left (2+x \right ) x^{7}-3179520 \ln \left (2+x \right ) x^{5}-5670144 \ln \left (2+x \right ) x^{4}+5666292 \ln \left (1+x \right ) x^{4}+6354720 \ln \left (1+x \right ) x^{3}-6359040 \ln \left (2+x \right ) x^{3}-4380672 \ln \left (2+x \right ) x^{2}+4377696 \ln \left (1+x \right ) x^{2}+1694592 \ln \left (1+x \right ) x}{12 \left (x^{2}+3 x +2\right )^{4}}\) | \(200\) |
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Time = 0.25 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.59 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=-\frac {17496 \, x^{7} + 184200 \, x^{6} + 813888 \, x^{5} + 1955853 \, x^{4} + 2759400 \, x^{3} + 2286008 \, x^{2} - 17664 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )} \log \left (x + 2\right ) + 17652 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )} \log \left (x + 1\right ) + 1030560 \, x + 195280}{12 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=\frac {- 17496 x^{7} - 184200 x^{6} - 813888 x^{5} - 1955853 x^{4} - 2759400 x^{3} - 2286008 x^{2} - 1030560 x - 195280}{12 x^{8} + 144 x^{7} + 744 x^{6} + 2160 x^{5} + 3852 x^{4} + 4320 x^{3} + 2976 x^{2} + 1152 x + 192} - 1471 \log {\left (x + 1 \right )} + 1472 \log {\left (x + 2 \right )} \]
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Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=-\frac {17496 \, x^{7} + 184200 \, x^{6} + 813888 \, x^{5} + 1955853 \, x^{4} + 2759400 \, x^{3} + 2286008 \, x^{2} + 1030560 \, x + 195280}{12 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )}} + 1472 \, \log \left (x + 2\right ) - 1471 \, \log \left (x + 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.60 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=-\frac {17496 \, x^{7} + 184200 \, x^{6} + 813888 \, x^{5} + 1955853 \, x^{4} + 2759400 \, x^{3} + 2286008 \, x^{2} + 1030560 \, x + 195280}{12 \, {\left (x + 2\right )}^{4} {\left (x + 1\right )}^{4}} + 1472 \, \log \left ({\left | x + 2 \right |}\right ) - 1471 \, \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=1472\,\ln \left (x+2\right )-1471\,\ln \left (x+1\right )-\frac {1458\,x^7+15350\,x^6+67824\,x^5+\frac {651951\,x^4}{4}+229950\,x^3+\frac {571502\,x^2}{3}+85880\,x+\frac {48820}{3}}{x^8+12\,x^7+62\,x^6+180\,x^5+321\,x^4+360\,x^3+248\,x^2+96\,x+16} \]
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