Integrand size = 14, antiderivative size = 104 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=735 x+\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {x^2 (2206+1593 x)}{2 \left (2+3 x+x^2\right )}-1471 \log (1+x)+1472 \log (2+x) \] Output:
735*x+1/4*x^8*(4+3*x)/(x^2+3*x+2)^4-1/12*x^6*(110+81*x)/(x^2+3*x+2)^3+1/2* x^4*(184+135*x)/(x^2+3*x+2)^2-1/2*x^2*(2206+1593*x)/(x^2+3*x+2)-1471*ln(1+ x)+1472*ln(2+x)
Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=\frac {514+513 x}{4 \left (2+3 x+x^2\right )^4}+\frac {415+1998 x}{12 \left (2+3 x+x^2\right )^3}+\frac {3 (451+456 x)}{4 \left (2+3 x+x^2\right )^2}-\frac {2 (1114+729 x)}{2+3 x+x^2}-1471 \log (1+x)+1472 \log (2+x) \] Input:
Integrate[x^9/(2 + 3*x + x^2)^5,x]
Output:
(514 + 513*x)/(4*(2 + 3*x + x^2)^4) + (415 + 1998*x)/(12*(2 + 3*x + x^2)^3 ) + (3*(451 + 456*x))/(4*(2 + 3*x + x^2)^2) - (2*(1114 + 729*x))/(2 + 3*x + x^2) - 1471*Log[1 + x] + 1472*Log[2 + x]
Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.72, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1141, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^9}{\left (x^2+3 x+2\right )^5} \, dx\) |
\(\Big \downarrow \) 1141 |
\(\displaystyle \int \left (\frac {1472}{x+2}+\frac {1024}{(x+2)^2}+\frac {768}{(x+2)^3}+\frac {256}{(x+2)^4}+\frac {512}{(x+2)^5}-\frac {1471}{x+1}+\frac {434}{(x+1)^2}-\frac {96}{(x+1)^3}+\frac {14}{(x+1)^4}-\frac {1}{(x+1)^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {434}{x+1}-\frac {1024}{x+2}+\frac {48}{(x+1)^2}-\frac {384}{(x+2)^2}-\frac {14}{3 (x+1)^3}-\frac {256}{3 (x+2)^3}+\frac {1}{4 (x+1)^4}-\frac {128}{(x+2)^4}-1471 \log (x+1)+1472 \log (x+2)\) |
Input:
Int[x^9/(2 + 3*x + x^2)^5,x]
Output:
1/(4*(1 + x)^4) - 14/(3*(1 + x)^3) + 48/(1 + x)^2 - 434/(1 + x) - 128/(2 + x)^4 - 256/(3*(2 + x)^3) - 384/(2 + x)^2 - 1024/(2 + x) - 1471*Log[1 + x] + 1472*Log[2 + x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[ (d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 1] || !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
Time = 0.13 (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 | \(-\frac {195280+1030560 x +17496 x^{7}+17652 \ln \left (1+x \right ) x^{8}-17664 \ln \left (2+x \right ) x^{8}+211824 \ln \left (1+x \right ) x^{7}-211968 \ln \left (2+x \right ) x^{7}+1094424 \ln \left (1+x \right ) x^{6}-1095168 \ln \left (2+x \right ) x^{6}+3177360 \ln \left (1+x \right ) x^{5}+5666292 \ln \left (1+x \right ) x^{4}+184200 x^{6}+282432 \ln \left (1+x \right )-282624 \ln \left (2+x \right )+1955853 x^{4}+2286008 x^{2}+2759400 x^{3}+1694592 \ln \left (1+x \right ) x +4377696 \ln \left (1+x \right ) x^{2}+813888 x^{5}-1695744 \ln \left (2+x \right ) x +6354720 \ln \left (1+x \right ) x^{3}-6359040 \ln \left (2+x \right ) x^{3}-4380672 \ln \left (2+x \right ) x^{2}-3179520 \ln \left (2+x \right ) x^{5}-5670144 \ln \left (2+x \right ) x^{4}}{12 \left (x^{2}+3 x +2\right )^{4}}\) | \(200\) |
Input:
int(x^9/(x^2+3*x+2)^5,x,method=_RETURNVERBOSE)
Output:
(-229950*x^3-85880*x-67824*x^5-15350*x^6-1458*x^7-651951/4*x^4-571502/3*x^ 2-48820/3)/(x^2+3*x+2)^4-1471*ln(1+x)+1472*ln(2+x)
Time = 0.06 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.59 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=-\frac {17496 \, x^{7} + 184200 \, x^{6} + 813888 \, x^{5} + 1955853 \, x^{4} + 2759400 \, x^{3} + 2286008 \, x^{2} - 17664 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )} \log \left (x + 2\right ) + 17652 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )} \log \left (x + 1\right ) + 1030560 \, x + 195280}{12 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )}} \] Input:
integrate(x^9/(x^2+3*x+2)^5,x, algorithm="fricas")
Output:
-1/12*(17496*x^7 + 184200*x^6 + 813888*x^5 + 1955853*x^4 + 2759400*x^3 + 2 286008*x^2 - 17664*(x^8 + 12*x^7 + 62*x^6 + 180*x^5 + 321*x^4 + 360*x^3 + 248*x^2 + 96*x + 16)*log(x + 2) + 17652*(x^8 + 12*x^7 + 62*x^6 + 180*x^5 + 321*x^4 + 360*x^3 + 248*x^2 + 96*x + 16)*log(x + 1) + 1030560*x + 195280) /(x^8 + 12*x^7 + 62*x^6 + 180*x^5 + 321*x^4 + 360*x^3 + 248*x^2 + 96*x + 1 6)
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=\frac {- 17496 x^{7} - 184200 x^{6} - 813888 x^{5} - 1955853 x^{4} - 2759400 x^{3} - 2286008 x^{2} - 1030560 x - 195280}{12 x^{8} + 144 x^{7} + 744 x^{6} + 2160 x^{5} + 3852 x^{4} + 4320 x^{3} + 2976 x^{2} + 1152 x + 192} - 1471 \log {\left (x + 1 \right )} + 1472 \log {\left (x + 2 \right )} \] Input:
integrate(x**9/(x**2+3*x+2)**5,x)
Output:
(-17496*x**7 - 184200*x**6 - 813888*x**5 - 1955853*x**4 - 2759400*x**3 - 2 286008*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(x + 1) + 1472*log(x + 2)
Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=-\frac {17496 \, x^{7} + 184200 \, x^{6} + 813888 \, x^{5} + 1955853 \, x^{4} + 2759400 \, x^{3} + 2286008 \, x^{2} + 1030560 \, x + 195280}{12 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )}} + 1472 \, \log \left (x + 2\right ) - 1471 \, \log \left (x + 1\right ) \] Input:
integrate(x^9/(x^2+3*x+2)^5,x, algorithm="maxima")
Output:
-1/12*(17496*x^7 + 184200*x^6 + 813888*x^5 + 1955853*x^4 + 2759400*x^3 + 2 286008*x^2 + 1030560*x + 195280)/(x^8 + 12*x^7 + 62*x^6 + 180*x^5 + 321*x^ 4 + 360*x^3 + 248*x^2 + 96*x + 16) + 1472*log(x + 2) - 1471*log(x + 1)
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.60 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=-\frac {17496 \, x^{7} + 184200 \, x^{6} + 813888 \, x^{5} + 1955853 \, x^{4} + 2759400 \, x^{3} + 2286008 \, x^{2} + 1030560 \, x + 195280}{12 \, {\left (x + 2\right )}^{4} {\left (x + 1\right )}^{4}} + 1472 \, \log \left ({\left | x + 2 \right |}\right ) - 1471 \, \log \left ({\left | x + 1 \right |}\right ) \] Input:
integrate(x^9/(x^2+3*x+2)^5,x, algorithm="giac")
Output:
-1/12*(17496*x^7 + 184200*x^6 + 813888*x^5 + 1955853*x^4 + 2759400*x^3 + 2 286008*x^2 + 1030560*x + 195280)/((x + 2)^4*(x + 1)^4) + 1472*log(abs(x + 2)) - 1471*log(abs(x + 1))
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=1472\,\ln \left (x+2\right )-1471\,\ln \left (x+1\right )-\frac {1458\,x^7+15350\,x^6+67824\,x^5+\frac {651951\,x^4}{4}+229950\,x^3+\frac {571502\,x^2}{3}+85880\,x+\frac {48820}{3}}{x^8+12\,x^7+62\,x^6+180\,x^5+321\,x^4+360\,x^3+248\,x^2+96\,x+16} \] Input:
int(x^9/(3*x + x^2 + 2)^5,x)
Output:
1472*log(x + 2) - 1471*log(x + 1) - (85880*x + (571502*x^2)/3 + 229950*x^3 + (651951*x^4)/4 + 67824*x^5 + 15350*x^6 + 1458*x^7 + 48820/3)/(96*x + 24 8*x^2 + 360*x^3 + 321*x^4 + 180*x^5 + 62*x^6 + 12*x^7 + x^8 + 16)
Time = 0.16 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.21 \[ \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx=\frac {-171952-890592 x -1924424 x^{2}-5666292 \,\mathrm {log}\left (x +1\right ) x^{4}-6354720 \,\mathrm {log}\left (x +1\right ) x^{3}-551448 x^{5}-1694592 \,\mathrm {log}\left (x +1\right ) x +282624 \,\mathrm {log}\left (x +2\right )+1695744 \,\mathrm {log}\left (x +2\right ) x -282432 \,\mathrm {log}\left (x +1\right )-2234520 x^{3}-93804 x^{6}+17664 \,\mathrm {log}\left (x +2\right ) x^{8}+211968 \,\mathrm {log}\left (x +2\right ) x^{7}+1095168 \,\mathrm {log}\left (x +2\right ) x^{6}-17652 \,\mathrm {log}\left (x +1\right ) x^{8}-211824 \,\mathrm {log}\left (x +1\right ) x^{7}-1094424 \,\mathrm {log}\left (x +1\right ) x^{6}-3177360 \,\mathrm {log}\left (x +1\right ) x^{5}+1458 x^{8}-1487835 x^{4}+3179520 \,\mathrm {log}\left (x +2\right ) x^{5}+5670144 \,\mathrm {log}\left (x +2\right ) x^{4}+6359040 \,\mathrm {log}\left (x +2\right ) x^{3}+4380672 \,\mathrm {log}\left (x +2\right ) x^{2}-4377696 \,\mathrm {log}\left (x +1\right ) x^{2}}{12 x^{8}+144 x^{7}+744 x^{6}+2160 x^{5}+3852 x^{4}+4320 x^{3}+2976 x^{2}+1152 x +192} \] Input:
int(x^9/(x^2+3*x+2)^5,x)
Output:
(17664*log(x + 2)*x**8 + 211968*log(x + 2)*x**7 + 1095168*log(x + 2)*x**6 + 3179520*log(x + 2)*x**5 + 5670144*log(x + 2)*x**4 + 6359040*log(x + 2)*x **3 + 4380672*log(x + 2)*x**2 + 1695744*log(x + 2)*x + 282624*log(x + 2) - 17652*log(x + 1)*x**8 - 211824*log(x + 1)*x**7 - 1094424*log(x + 1)*x**6 - 3177360*log(x + 1)*x**5 - 5666292*log(x + 1)*x**4 - 6354720*log(x + 1)*x **3 - 4377696*log(x + 1)*x**2 - 1694592*log(x + 1)*x - 282432*log(x + 1) + 1458*x**8 - 93804*x**6 - 551448*x**5 - 1487835*x**4 - 2234520*x**3 - 1924 424*x**2 - 890592*x - 171952)/(12*(x**8 + 12*x**7 + 62*x**6 + 180*x**5 + 3 21*x**4 + 360*x**3 + 248*x**2 + 96*x + 16))