Integrand size = 22, antiderivative size = 81 \[ \int \frac {(2+3 x)^8 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {697540921}{2048 (1-2 x)}+\frac {2330515357 x}{1024}+\frac {413355417 x^2}{256}+\frac {346239417 x^3}{256}+\frac {275757561 x^4}{256}+\frac {235268793 x^5}{320}+396738 x^6+\frac {17378631 x^7}{112}+\frac {1235655 x^8}{32}+\frac {18225 x^9}{4}+\frac {1512848491 \log (1-2 x)}{1024} \]
697540921/2048/(1-2*x)+2330515357/1024*x+413355417/256*x^2+346239417/256*x ^3+275757561/256*x^4+235268793/320*x^5+396738*x^6+17378631/112*x^7+1235655 /32*x^8+18225/4*x^9+1512848491/1024*ln(1-2*x)
Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^8 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {420890769939-1689637297718 x+842130532880 x^2+538127987040 x^3+466727825760 x^4+406896098112 x^5+307848957696 x^6+183016143360 x^7+77907121920 x^8+20836569600 x^9+2612736000 x^{10}+423597577480 (-1+2 x) \log (1-2 x)}{286720 (-1+2 x)} \]
(420890769939 - 1689637297718*x + 842130532880*x^2 + 538127987040*x^3 + 46 6727825760*x^4 + 406896098112*x^5 + 307848957696*x^6 + 183016143360*x^7 + 77907121920*x^8 + 20836569600*x^9 + 2612736000*x^10 + 423597577480*(-1 + 2 *x)*Log[1 - 2*x])/(286720*(-1 + 2*x))
Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {99, 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.
| \(\displaystyle \int \frac {(3 x+2)^8 (5 x+3)^2}{(1-2 x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {164025 x^8}{4}+\frac {1235655 x^7}{4}+\frac {17378631 x^6}{16}+2380428 x^5+\frac {235268793 x^4}{64}+\frac {275757561 x^3}{64}+\frac {1038718251 x^2}{256}+\frac {413355417 x}{128}+\frac {1512848491}{512 (2 x-1)}+\frac {697540921}{1024 (2 x-1)^2}+\frac {2330515357}{1024}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {18225 x^9}{4}+\frac {1235655 x^8}{32}+\frac {17378631 x^7}{112}+396738 x^6+\frac {235268793 x^5}{320}+\frac {275757561 x^4}{256}+\frac {346239417 x^3}{256}+\frac {413355417 x^2}{256}+\frac {2330515357 x}{1024}+\frac {697540921}{2048 (1-2 x)}+\frac {1512848491 \log (1-2 x)}{1024}\) |
697540921/(2048*(1 - 2*x)) + (2330515357*x)/1024 + (413355417*x^2)/256 + ( 346239417*x^3)/256 + (275757561*x^4)/256 + (235268793*x^5)/320 + 396738*x^ 6 + (17378631*x^7)/112 + (1235655*x^8)/32 + (18225*x^9)/4 + (1512848491*Lo g[1 - 2*x])/1024
3.16.54.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 2.64 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(\frac {18225 x^{9}}{4}+\frac {1235655 x^{8}}{32}+\frac {17378631 x^{7}}{112}+396738 x^{6}+\frac {235268793 x^{5}}{320}+\frac {275757561 x^{4}}{256}+\frac {346239417 x^{3}}{256}+\frac {413355417 x^{2}}{256}+\frac {2330515357 x}{1024}-\frac {697540921}{4096 \left (x -\frac {1}{2}\right )}+\frac {1512848491 \ln \left (-1+2 x \right )}{1024}\) | \(60\) |
| default | \(\frac {18225 x^{9}}{4}+\frac {1235655 x^{8}}{32}+\frac {17378631 x^{7}}{112}+396738 x^{6}+\frac {235268793 x^{5}}{320}+\frac {275757561 x^{4}}{256}+\frac {346239417 x^{3}}{256}+\frac {413355417 x^{2}}{256}+\frac {2330515357 x}{1024}+\frac {1512848491 \ln \left (-1+2 x \right )}{1024}-\frac {697540921}{2048 \left (-1+2 x \right )}\) | \(62\) |
| norman | \(\frac {-\frac {1514028139}{512} x +\frac {1503804523}{512} x^{2}+\frac {480471417}{256} x^{3}+\frac {416721273}{256} x^{4}+\frac {908250219}{640} x^{5}+\frac {171790713}{160} x^{6}+\frac {71490681}{112} x^{7}+\frac {60864939}{224} x^{8}+\frac {1162755}{16} x^{9}+\frac {18225}{2} x^{10}}{-1+2 x}+\frac {1512848491 \ln \left (-1+2 x \right )}{1024}\) | \(67\) |
| parallelrisch | \(\frac {326592000 x^{10}+2604571200 x^{9}+9738390240 x^{8}+22877017920 x^{7}+38481119712 x^{6}+50862012264 x^{5}+58340978220 x^{4}+67265998380 x^{3}+105899394370 \ln \left (x -\frac {1}{2}\right ) x +105266316610 x^{2}-52949697185 \ln \left (x -\frac {1}{2}\right )-105981969730 x}{-35840+71680 x}\) | \(72\) |
| meijerg | \(\frac {19968 x}{1-2 x}+\frac {114291 x \left (-1280 x^{6}-896 x^{5}-672 x^{4}-560 x^{3}-560 x^{2}-840 x +840\right )}{320 \left (1-2 x \right )}+\frac {1512848491 \ln \left (1-2 x \right )}{1024}+\frac {350001 x \left (-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{8960 \left (1-2 x \right )}+\frac {60928 x \left (-6 x +6\right )}{3 \left (1-2 x \right )}+\frac {31128 x \left (-8 x^{2}-12 x +12\right )}{1-2 x}+\frac {11130 x \left (-40 x^{3}-40 x^{2}-60 x +60\right )}{1-2 x}+\frac {15309 x \left (-8960 x^{8}-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{2048 \left (1-2 x \right )}+\frac {18225 x \left (-157696 x^{9}-98560 x^{8}-63360 x^{7}-42240 x^{6}-29568 x^{5}-22176 x^{4}-18480 x^{3}-18480 x^{2}-27720 x +27720\right )}{315392 \left (1-2 x \right )}+\frac {12789 x \left (-48 x^{4}-40 x^{3}-40 x^{2}-60 x +60\right )}{1-2 x}+\frac {55971 x \left (-448 x^{5}-336 x^{4}-280 x^{3}-280 x^{2}-420 x +420\right )}{40 \left (1-2 x \right )}\) | \(335\) |
18225/4*x^9+1235655/32*x^8+17378631/112*x^7+396738*x^6+235268793/320*x^5+2 75757561/256*x^4+346239417/256*x^3+413355417/256*x^2+2330515357/1024*x-697 540921/4096/(x-1/2)+1512848491/1024*ln(-1+2*x)
Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^8 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {653184000 \, x^{10} + 5209142400 \, x^{9} + 19476780480 \, x^{8} + 45754035840 \, x^{7} + 76962239424 \, x^{6} + 101724024528 \, x^{5} + 116681956440 \, x^{4} + 134531996760 \, x^{3} + 210532633220 \, x^{2} + 105899394370 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 163136074990 \, x - 24413932235}{71680 \, {\left (2 \, x - 1\right )}} \]
1/71680*(653184000*x^10 + 5209142400*x^9 + 19476780480*x^8 + 45754035840*x ^7 + 76962239424*x^6 + 101724024528*x^5 + 116681956440*x^4 + 134531996760* x^3 + 210532633220*x^2 + 105899394370*(2*x - 1)*log(2*x - 1) - 16313607499 0*x - 24413932235)/(2*x - 1)
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^8 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {18225 x^{9}}{4} + \frac {1235655 x^{8}}{32} + \frac {17378631 x^{7}}{112} + 396738 x^{6} + \frac {235268793 x^{5}}{320} + \frac {275757561 x^{4}}{256} + \frac {346239417 x^{3}}{256} + \frac {413355417 x^{2}}{256} + \frac {2330515357 x}{1024} + \frac {1512848491 \log {\left (2 x - 1 \right )}}{1024} - \frac {697540921}{4096 x - 2048} \]
18225*x**9/4 + 1235655*x**8/32 + 17378631*x**7/112 + 396738*x**6 + 2352687 93*x**5/320 + 275757561*x**4/256 + 346239417*x**3/256 + 413355417*x**2/256 + 2330515357*x/1024 + 1512848491*log(2*x - 1)/1024 - 697540921/(4096*x - 2048)
Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^8 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {18225}{4} \, x^{9} + \frac {1235655}{32} \, x^{8} + \frac {17378631}{112} \, x^{7} + 396738 \, x^{6} + \frac {235268793}{320} \, x^{5} + \frac {275757561}{256} \, x^{4} + \frac {346239417}{256} \, x^{3} + \frac {413355417}{256} \, x^{2} + \frac {2330515357}{1024} \, x - \frac {697540921}{2048 \, {\left (2 \, x - 1\right )}} + \frac {1512848491}{1024} \, \log \left (2 \, x - 1\right ) \]
18225/4*x^9 + 1235655/32*x^8 + 17378631/112*x^7 + 396738*x^6 + 235268793/3 20*x^5 + 275757561/256*x^4 + 346239417/256*x^3 + 413355417/256*x^2 + 23305 15357/1024*x - 697540921/2048/(2*x - 1) + 1512848491/1024*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.37 \[ \int \frac {(2+3 x)^8 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {1}{286720} \, {\left (2 \, x - 1\right )}^{9} {\left (\frac {66211425}{2 \, x - 1} + \frac {785410020}{{\left (2 \, x - 1\right )}^{2}} + \frac {5635662480}{{\left (2 \, x - 1\right )}^{3}} + \frac {27294241464}{{\left (2 \, x - 1\right )}^{4}} + \frac {94415339340}{{\left (2 \, x - 1\right )}^{5}} + \frac {241909873800}{{\left (2 \, x - 1\right )}^{6}} + \frac {478116124080}{{\left (2 \, x - 1\right )}^{7}} + \frac {826787759420}{{\left (2 \, x - 1\right )}^{8}} + 2551500\right )} - \frac {697540921}{2048 \, {\left (2 \, x - 1\right )}} - \frac {1512848491}{1024} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) \]
1/286720*(2*x - 1)^9*(66211425/(2*x - 1) + 785410020/(2*x - 1)^2 + 5635662 480/(2*x - 1)^3 + 27294241464/(2*x - 1)^4 + 94415339340/(2*x - 1)^5 + 2419 09873800/(2*x - 1)^6 + 478116124080/(2*x - 1)^7 + 826787759420/(2*x - 1)^8 + 2551500) - 697540921/2048/(2*x - 1) - 1512848491/1024*log(1/2*abs(2*x - 1)/(2*x - 1)^2)
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.73 \[ \int \frac {(2+3 x)^8 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {2330515357\,x}{1024}+\frac {1512848491\,\ln \left (x-\frac {1}{2}\right )}{1024}-\frac {697540921}{4096\,\left (x-\frac {1}{2}\right )}+\frac {413355417\,x^2}{256}+\frac {346239417\,x^3}{256}+\frac {275757561\,x^4}{256}+\frac {235268793\,x^5}{320}+396738\,x^6+\frac {17378631\,x^7}{112}+\frac {1235655\,x^8}{32}+\frac {18225\,x^9}{4} \]