Integrand size = 25, antiderivative size = 81 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^3 \left (2+3 x+x^2\right )} \, dx=\frac {8}{766521 (5-2 x)^2}+\frac {14960}{1110688929 (5-2 x)}+\frac {81}{48668 (4+3 x)^2}+\frac {2835}{1119364 (4+3 x)}-\frac {8401264 \log (5-2 x)}{1609388258121}+\frac {1}{343} \log (1+x)+\frac {\log (2+x)}{5832}-\frac {158679 \log (4+3 x)}{51490744} \] Output:
8/766521/(5-2*x)^2+14960/(5553444645-2221377858*x)+81/48668/(4+3*x)^2+2835 /(4477456+3358092*x)-8401264/1609388258121*ln(5-2*x)+1/343*ln(1+x)+1/5832* ln(2+x)-158679/51490744*ln(4+3*x)
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^3 \left (2+3 x+x^2\right )} \, dx=\frac {8}{766521 (5-2 x)^2}+\frac {14960}{1110688929 (5-2 x)}+\frac {81}{48668 (4+3 x)^2}+\frac {2835}{1119364 (4+3 x)}-\frac {8401264 \log (5-2 x)}{1609388258121}+\frac {1}{343} \log (2 (1+x))+\frac {\log (2 (2+x))}{5832}-\frac {158679 \log (8+6 x)}{51490744} \] Input:
Integrate[1/((5 - 2*x)^3*(4 + 3*x)^3*(2 + 3*x + x^2)),x]
Output:
8/(766521*(5 - 2*x)^2) + 14960/(1110688929*(5 - 2*x)) + 81/(48668*(4 + 3*x )^2) + 2835/(1119364*(4 + 3*x)) - (8401264*Log[5 - 2*x])/1609388258121 + L og[2*(1 + x)]/343 + Log[2*(2 + x)]/5832 - (158679*Log[8 + 6*x])/51490744
Time = 0.39 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1200, 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.
| \(\displaystyle \int \frac {1}{(5-2 x)^3 (3 x+4)^3 \left (x^2+3 x+2\right )} \, dx\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \int \left (\frac {1}{5832 (x+2)}-\frac {16802528}{1609388258121 (2 x-5)}-\frac {476037}{51490744 (3 x+4)}+\frac {29920}{1110688929 (2 x-5)^2}-\frac {8505}{1119364 (3 x+4)^2}-\frac {32}{766521 (2 x-5)^3}-\frac {243}{24334 (3 x+4)^3}+\frac {1}{343 (x+1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {14960}{1110688929 (5-2 x)}+\frac {2835}{1119364 (3 x+4)}+\frac {8}{766521 (5-2 x)^2}+\frac {81}{48668 (3 x+4)^2}-\frac {8401264 \log (5-2 x)}{1609388258121}+\frac {1}{343} \log (x+1)+\frac {\log (x+2)}{5832}-\frac {158679 \log (3 x+4)}{51490744}\) |
Input:
Int[1/((5 - 2*x)^3*(4 + 3*x)^3*(2 + 3*x + x^2)),x]
Output:
8/(766521*(5 - 2*x)^2) + 14960/(1110688929*(5 - 2*x)) + 81/(48668*(4 + 3*x )^2) + 2835/(1119364*(4 + 3*x)) - (8401264*Log[5 - 2*x])/1609388258121 + L og[1 + x]/343 + Log[2 + x]/5832 - (158679*Log[4 + 3*x])/51490744
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Time = 1.82 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74
| method | result | size |
| norman | \(\frac {-\frac {197766763}{4442755716} x -\frac {38773190}{370229643} x^{2}+\frac {3720785}{123409881} x^{3}+\frac {1315596763}{4442755716}}{\left (-5+2 x \right )^{2} \left (3 x +4\right )^{2}}-\frac {8401264 \ln \left (-5+2 x \right )}{1609388258121}+\frac {\ln \left (x +1\right )}{343}+\frac {\ln \left (2+x \right )}{5832}-\frac {158679 \ln \left (3 x +4\right )}{51490744}\) | \(60\) |
| risch | \(\frac {-\frac {197766763}{4442755716} x -\frac {38773190}{370229643} x^{2}+\frac {3720785}{123409881} x^{3}+\frac {1315596763}{4442755716}}{\left (-5+2 x \right )^{2} \left (3 x +4\right )^{2}}-\frac {8401264 \ln \left (-5+2 x \right )}{1609388258121}+\frac {\ln \left (x +1\right )}{343}+\frac {\ln \left (2+x \right )}{5832}-\frac {158679 \ln \left (3 x +4\right )}{51490744}\) | \(61\) |
| default | \(\frac {8}{766521 \left (-5+2 x \right )^{2}}-\frac {14960}{1110688929 \left (-5+2 x \right )}-\frac {8401264 \ln \left (-5+2 x \right )}{1609388258121}+\frac {\ln \left (x +1\right )}{343}+\frac {81}{48668 \left (3 x +4\right )^{2}}+\frac {2835}{1119364 \left (3 x +4\right )}-\frac {158679 \ln \left (3 x +4\right )}{51490744}+\frac {\ln \left (2+x \right )}{5832}\) | \(66\) |
| parallelrisch | \(\frac {-573128079174 x +388182057480 x^{3}-7169519703816 \ln \left (x +1\right ) x^{2}-1348376455440 x^{2}+3812599419174+10510290665280 \ln \left (x +1\right ) x -26884044800 \ln \left (x -\frac {5}{2}\right )-15870883165200 \ln \left (x +\frac {4}{3}\right )+15014700950400 \ln \left (x +1\right )-11109618215640 \ln \left (x +\frac {4}{3}\right ) x -421664138959 \ln \left (2+x \right ) x^{2}+618146381720 \ln \left (2+x \right ) x +883066259600 \ln \left (2+x \right )-1428379484868 \ln \left (x +\frac {4}{3}\right ) x^{4}-185443914516 \ln \left (2+x \right ) x^{3}-2419564032 \ln \left (x -\frac {5}{2}\right ) x^{4}+12837131392 \ln \left (x -\frac {5}{2}\right ) x^{2}-18818831360 \ln \left (x -\frac {5}{2}\right ) x +7578346711383 \ln \left (x +\frac {4}{3}\right ) x^{2}+5645649408 \ln \left (x -\frac {5}{2}\right ) x^{3}-3153087199584 \ln \left (x +1\right ) x^{3}+79475963364 \ln \left (2+x \right ) x^{4}+3332885464692 \ln \left (x +\frac {4}{3}\right ) x^{3}+1351323085536 \ln \left (x +1\right ) x^{4}}{12875106064968 \left (-5+2 x \right )^{2} \left (3 x +4\right )^{2}}\) | \(192\) |
Input:
int(1/(5-2*x)^3/(3*x+4)^3/(x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
(-197766763/4442755716*x-38773190/370229643*x^2+3720785/123409881*x^3+1315 596763/4442755716)/(-5+2*x)^2/(3*x+4)^2-8401264/1609388258121*ln(-5+2*x)+1 /343*ln(x+1)+1/5832*ln(2+x)-158679/51490744*ln(3*x+4)
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (65) = 130\).
Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.81 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^3 \left (2+3 x+x^2\right )} \, dx=\frac {388182057480 \, x^{3} - 1348376455440 \, x^{2} - 39677207913 \, {\left (36 \, x^{4} - 84 \, x^{3} - 191 \, x^{2} + 280 \, x + 400\right )} \log \left (3 \, x + 4\right ) - 67210112 \, {\left (36 \, x^{4} - 84 \, x^{3} - 191 \, x^{2} + 280 \, x + 400\right )} \log \left (2 \, x - 5\right ) + 2207665649 \, {\left (36 \, x^{4} - 84 \, x^{3} - 191 \, x^{2} + 280 \, x + 400\right )} \log \left (x + 2\right ) + 37536752376 \, {\left (36 \, x^{4} - 84 \, x^{3} - 191 \, x^{2} + 280 \, x + 400\right )} \log \left (x + 1\right ) - 573128079174 \, x + 3812599419174}{12875106064968 \, {\left (36 \, x^{4} - 84 \, x^{3} - 191 \, x^{2} + 280 \, x + 400\right )}} \] Input:
integrate(1/(5-2*x)^3/(4+3*x)^3/(x^2+3*x+2),x, algorithm="fricas")
Output:
1/12875106064968*(388182057480*x^3 - 1348376455440*x^2 - 39677207913*(36*x ^4 - 84*x^3 - 191*x^2 + 280*x + 400)*log(3*x + 4) - 67210112*(36*x^4 - 84* x^3 - 191*x^2 + 280*x + 400)*log(2*x - 5) + 2207665649*(36*x^4 - 84*x^3 - 191*x^2 + 280*x + 400)*log(x + 2) + 37536752376*(36*x^4 - 84*x^3 - 191*x^2 + 280*x + 400)*log(x + 1) - 573128079174*x + 3812599419174)/(36*x^4 - 84* x^3 - 191*x^2 + 280*x + 400)
Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^3 \left (2+3 x+x^2\right )} \, dx=- \frac {- 133948260 x^{3} + 465278280 x^{2} + 197766763 x - 1315596763}{159939205776 x^{4} - 373191480144 x^{3} - 848566341756 x^{2} + 1243971600480 x + 1777102286400} - \frac {8401264 \log {\left (x - \frac {5}{2} \right )}}{1609388258121} + \frac {\log {\left (x + 1 \right )}}{343} - \frac {158679 \log {\left (x + \frac {4}{3} \right )}}{51490744} + \frac {\log {\left (x + 2 \right )}}{5832} \] Input:
integrate(1/(5-2*x)**3/(4+3*x)**3/(x**2+3*x+2),x)
Output:
-(-133948260*x**3 + 465278280*x**2 + 197766763*x - 1315596763)/(1599392057 76*x**4 - 373191480144*x**3 - 848566341756*x**2 + 1243971600480*x + 177710 2286400) - 8401264*log(x - 5/2)/1609388258121 + log(x + 1)/343 - 158679*lo g(x + 4/3)/51490744 + log(x + 2)/5832
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^3 \left (2+3 x+x^2\right )} \, dx=\frac {133948260 \, x^{3} - 465278280 \, x^{2} - 197766763 \, x + 1315596763}{4442755716 \, {\left (36 \, x^{4} - 84 \, x^{3} - 191 \, x^{2} + 280 \, x + 400\right )}} - \frac {158679}{51490744} \, \log \left (3 \, x + 4\right ) - \frac {8401264}{1609388258121} \, \log \left (2 \, x - 5\right ) + \frac {1}{5832} \, \log \left (x + 2\right ) + \frac {1}{343} \, \log \left (x + 1\right ) \] Input:
integrate(1/(5-2*x)^3/(4+3*x)^3/(x^2+3*x+2),x, algorithm="maxima")
Output:
1/4442755716*(133948260*x^3 - 465278280*x^2 - 197766763*x + 1315596763)/(3 6*x^4 - 84*x^3 - 191*x^2 + 280*x + 400) - 158679/51490744*log(3*x + 4) - 8 401264/1609388258121*log(2*x - 5) + 1/5832*log(x + 2) + 1/343*log(x + 1)
Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^3 \left (2+3 x+x^2\right )} \, dx=\frac {133948260 \, x^{3} - 465278280 \, x^{2} - 197766763 \, x + 1315596763}{4442755716 \, {\left (3 \, x + 4\right )}^{2} {\left (2 \, x - 5\right )}^{2}} - \frac {158679}{51490744} \, \log \left ({\left | 3 \, x + 4 \right |}\right ) - \frac {8401264}{1609388258121} \, \log \left ({\left | 2 \, x - 5 \right |}\right ) + \frac {1}{5832} \, \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{343} \, \log \left ({\left | x + 1 \right |}\right ) \] Input:
integrate(1/(5-2*x)^3/(4+3*x)^3/(x^2+3*x+2),x, algorithm="giac")
Output:
1/4442755716*(133948260*x^3 - 465278280*x^2 - 197766763*x + 1315596763)/(( 3*x + 4)^2*(2*x - 5)^2) - 158679/51490744*log(abs(3*x + 4)) - 8401264/1609 388258121*log(abs(2*x - 5)) + 1/5832*log(abs(x + 2)) + 1/343*log(abs(x + 1 ))
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^3 \left (2+3 x+x^2\right )} \, dx=\frac {\ln \left (x+1\right )}{343}+\frac {\ln \left (x+2\right )}{5832}-\frac {8401264\,\ln \left (x-\frac {5}{2}\right )}{1609388258121}-\frac {158679\,\ln \left (x+\frac {4}{3}\right )}{51490744}-\frac {-\frac {3720785\,x^3}{4442755716}+\frac {19386595\,x^2}{6664133574}+\frac {197766763\,x}{159939205776}-\frac {1315596763}{159939205776}}{x^4-\frac {7\,x^3}{3}-\frac {191\,x^2}{36}+\frac {70\,x}{9}+\frac {100}{9}} \] Input:
int(-1/((2*x - 5)^3*(3*x + 4)^3*(3*x + x^2 + 2)),x)
Output:
log(x + 1)/343 + log(x + 2)/5832 - (8401264*log(x - 5/2))/1609388258121 - (158679*log(x + 4/3))/51490744 - ((197766763*x)/159939205776 + (19386595*x ^2)/6664133574 - (3720785*x^3)/4442755716 - 1315596763/159939205776)/((70* x)/9 - (191*x^2)/36 - (7*x^3)/3 + x^4 + 100/9)
Time = 0.18 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.69 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^3 \left (2+3 x+x^2\right )} \, dx=\frac {-1428379484868 \,\mathrm {log}\left (3 x +4\right ) x^{4}+3332885464692 \,\mathrm {log}\left (3 x +4\right ) x^{3}+7578346711383 \,\mathrm {log}\left (3 x +4\right ) x^{2}-11109618215640 \,\mathrm {log}\left (3 x +4\right ) x -15870883165200 \,\mathrm {log}\left (3 x +4\right )-2419564032 \,\mathrm {log}\left (2 x -5\right ) x^{4}+5645649408 \,\mathrm {log}\left (2 x -5\right ) x^{3}+12837131392 \,\mathrm {log}\left (2 x -5\right ) x^{2}-18818831360 \,\mathrm {log}\left (2 x -5\right ) x -26884044800 \,\mathrm {log}\left (2 x -5\right )+79475963364 \,\mathrm {log}\left (x +2\right ) x^{4}-185443914516 \,\mathrm {log}\left (x +2\right ) x^{3}-421664138959 \,\mathrm {log}\left (x +2\right ) x^{2}+618146381720 \,\mathrm {log}\left (x +2\right ) x +883066259600 \,\mathrm {log}\left (x +2\right )+1351323085536 \,\mathrm {log}\left (x +1\right ) x^{4}-3153087199584 \,\mathrm {log}\left (x +1\right ) x^{3}-7169519703816 \,\mathrm {log}\left (x +1\right ) x^{2}+10510290665280 \,\mathrm {log}\left (x +1\right ) x +15014700950400 \,\mathrm {log}\left (x +1\right )+166363738920 x^{4}-2231028514710 x^{2}+720812112426 x +5661085407174}{463503818338848 x^{4}-1081508909457312 x^{3}-2459145258408888 x^{2}+3605029698191040 x +5150042425987200} \] Input:
int(1/(5-2*x)^3/(4+3*x)^3/(x^2+3*x+2),x)
Output:
( - 1428379484868*log(3*x + 4)*x**4 + 3332885464692*log(3*x + 4)*x**3 + 75 78346711383*log(3*x + 4)*x**2 - 11109618215640*log(3*x + 4)*x - 1587088316 5200*log(3*x + 4) - 2419564032*log(2*x - 5)*x**4 + 5645649408*log(2*x - 5) *x**3 + 12837131392*log(2*x - 5)*x**2 - 18818831360*log(2*x - 5)*x - 26884 044800*log(2*x - 5) + 79475963364*log(x + 2)*x**4 - 185443914516*log(x + 2 )*x**3 - 421664138959*log(x + 2)*x**2 + 618146381720*log(x + 2)*x + 883066 259600*log(x + 2) + 1351323085536*log(x + 1)*x**4 - 3153087199584*log(x + 1)*x**3 - 7169519703816*log(x + 1)*x**2 + 10510290665280*log(x + 1)*x + 15 014700950400*log(x + 1) + 166363738920*x**4 - 2231028514710*x**2 + 7208121 12426*x + 5661085407174)/(12875106064968*(36*x**4 - 84*x**3 - 191*x**2 + 2 80*x + 400))