Integrand size = 22, antiderivative size = 76 \[ \int \frac {(2+3 x)^7 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {99648703}{1024 (1-2 x)}+\frac {9077405 x}{16}+\frac {195497697 x^2}{512}+\frac {18842715 x^3}{64}+\frac {53086563 x^4}{256}+\frac {4750569 x^5}{40}+\frac {1597239 x^6}{32}+\frac {375435 x^7}{28}+\frac {54675 x^8}{32}+\frac {389535839 \log (1-2 x)}{1024} \]
99648703/1024/(1-2*x)+9077405/16*x+195497697/512*x^2+18842715/64*x^3+53086 563/256*x^4+4750569/40*x^5+1597239/32*x^6+375435/28*x^7+54675/32*x^8+38953 5839/1024*ln(1-2*x)
Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^7 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {96389258691-411248888662 x+215855484880 x^2+134542057440 x^3+109373775840 x^4+84861822528 x^5+53792895744 x^6+24778068480 x^7+7199020800 x^8+979776000 x^9+109070034920 (-1+2 x) \log (1-2 x)}{286720 (-1+2 x)} \]
(96389258691 - 411248888662*x + 215855484880*x^2 + 134542057440*x^3 + 1093 73775840*x^4 + 84861822528*x^5 + 53792895744*x^6 + 24778068480*x^7 + 71990 20800*x^8 + 979776000*x^9 + 109070034920*(-1 + 2*x)*Log[1 - 2*x])/(286720* (-1 + 2*x))
Time = 0.20 (sec) , antiderivative size = 76, 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.
| \(\displaystyle \int \frac {(3 x+2)^7 (5 x+3)^2}{(1-2 x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {54675 x^7}{4}+\frac {375435 x^6}{4}+\frac {4791717 x^5}{16}+\frac {4750569 x^4}{8}+\frac {53086563 x^3}{64}+\frac {56528145 x^2}{64}+\frac {195497697 x}{256}+\frac {389535839}{512 (2 x-1)}+\frac {99648703}{512 (2 x-1)^2}+\frac {9077405}{16}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {54675 x^8}{32}+\frac {375435 x^7}{28}+\frac {1597239 x^6}{32}+\frac {4750569 x^5}{40}+\frac {53086563 x^4}{256}+\frac {18842715 x^3}{64}+\frac {195497697 x^2}{512}+\frac {9077405 x}{16}+\frac {99648703}{1024 (1-2 x)}+\frac {389535839 \log (1-2 x)}{1024}\) |
99648703/(1024*(1 - 2*x)) + (9077405*x)/16 + (195497697*x^2)/512 + (188427 15*x^3)/64 + (53086563*x^4)/256 + (4750569*x^5)/40 + (1597239*x^6)/32 + (3 75435*x^7)/28 + (54675*x^8)/32 + (389535839*Log[1 - 2*x])/1024
3.16.55.3.1 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 = 0.86 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {54675 x^{8}}{32}+\frac {375435 x^{7}}{28}+\frac {1597239 x^{6}}{32}+\frac {4750569 x^{5}}{40}+\frac {53086563 x^{4}}{256}+\frac {18842715 x^{3}}{64}+\frac {195497697 x^{2}}{512}+\frac {9077405 x}{16}-\frac {99648703}{2048 \left (x -\frac {1}{2}\right )}+\frac {389535839 \ln \left (-1+2 x \right )}{1024}\) | \(55\) |
| default | \(\frac {54675 x^{8}}{32}+\frac {375435 x^{7}}{28}+\frac {1597239 x^{6}}{32}+\frac {4750569 x^{5}}{40}+\frac {53086563 x^{4}}{256}+\frac {18842715 x^{3}}{64}+\frac {195497697 x^{2}}{512}+\frac {9077405 x}{16}+\frac {389535839 \ln \left (-1+2 x \right )}{1024}-\frac {99648703}{1024 \left (-1+2 x \right )}\) | \(57\) |
| norman | \(\frac {-\frac {390125663}{512} x +\frac {385456223}{512} x^{2}+\frac {120126837}{256} x^{3}+\frac {97655157}{256} x^{4}+\frac {189423711}{640} x^{5}+\frac {30018357}{160} x^{6}+\frac {9678933}{112} x^{7}+\frac {5624235}{224} x^{8}+\frac {54675}{16} x^{9}}{-1+2 x}+\frac {389535839 \ln \left (-1+2 x \right )}{1024}\) | \(62\) |
| parallelrisch | \(\frac {122472000 x^{9}+899877600 x^{8}+3097258560 x^{7}+6724111968 x^{6}+10607727816 x^{5}+13671721980 x^{4}+16817757180 x^{3}+27267508730 \ln \left (x -\frac {1}{2}\right ) x +26981935610 x^{2}-13633754365 \ln \left (x -\frac {1}{2}\right )-27308796410 x}{-35840+71680 x}\) | \(67\) |
| meijerg | \(\frac {9120 x}{1-2 x}+\frac {389535839 \ln \left (1-2 x \right )}{1024}+\frac {24488 x \left (-6 x +6\right )}{3 \left (1-2 x \right )}+\frac {21945 x \left (-8 x^{2}-12 x +12\right )}{2 \left (1-2 x \right )}+\frac {6741 x \left (-40 x^{3}-40 x^{2}-60 x +60\right )}{2 \left (1-2 x \right )}+\frac {103509 x \left (-48 x^{4}-40 x^{3}-40 x^{2}-60 x +60\right )}{32 \left (1-2 x \right )}+\frac {90801 x \left (-448 x^{5}-336 x^{4}-280 x^{3}-280 x^{2}-420 x +420\right )}{320 \left (1-2 x \right )}+\frac {278721 x \left (-1280 x^{6}-896 x^{5}-672 x^{4}-560 x^{3}-560 x^{2}-840 x +840\right )}{5120 \left (1-2 x \right )}+\frac {891 x \left (-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{224 \left (1-2 x \right )}+\frac {10935 x \left (-8960 x^{8}-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{28672 \left (1-2 x \right )}\) | \(280\) |
54675/32*x^8+375435/28*x^7+1597239/32*x^6+4750569/40*x^5+53086563/256*x^4+ 18842715/64*x^3+195497697/512*x^2+9077405/16*x-99648703/2048/(x-1/2)+38953 5839/1024*ln(-1+2*x)
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^7 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {122472000 \, x^{9} + 899877600 \, x^{8} + 3097258560 \, x^{7} + 6724111968 \, x^{6} + 10607727816 \, x^{5} + 13671721980 \, x^{4} + 16817757180 \, x^{3} + 26981935610 \, x^{2} + 13633754365 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 20333387200 \, x - 3487704605}{35840 \, {\left (2 \, x - 1\right )}} \]
1/35840*(122472000*x^9 + 899877600*x^8 + 3097258560*x^7 + 6724111968*x^6 + 10607727816*x^5 + 13671721980*x^4 + 16817757180*x^3 + 26981935610*x^2 + 1 3633754365*(2*x - 1)*log(2*x - 1) - 20333387200*x - 3487704605)/(2*x - 1)
Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^7 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {54675 x^{8}}{32} + \frac {375435 x^{7}}{28} + \frac {1597239 x^{6}}{32} + \frac {4750569 x^{5}}{40} + \frac {53086563 x^{4}}{256} + \frac {18842715 x^{3}}{64} + \frac {195497697 x^{2}}{512} + \frac {9077405 x}{16} + \frac {389535839 \log {\left (2 x - 1 \right )}}{1024} - \frac {99648703}{2048 x - 1024} \]
54675*x**8/32 + 375435*x**7/28 + 1597239*x**6/32 + 4750569*x**5/40 + 53086 563*x**4/256 + 18842715*x**3/64 + 195497697*x**2/512 + 9077405*x/16 + 3895 35839*log(2*x - 1)/1024 - 99648703/(2048*x - 1024)
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^7 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {54675}{32} \, x^{8} + \frac {375435}{28} \, x^{7} + \frac {1597239}{32} \, x^{6} + \frac {4750569}{40} \, x^{5} + \frac {53086563}{256} \, x^{4} + \frac {18842715}{64} \, x^{3} + \frac {195497697}{512} \, x^{2} + \frac {9077405}{16} \, x - \frac {99648703}{1024 \, {\left (2 \, x - 1\right )}} + \frac {389535839}{1024} \, \log \left (2 \, x - 1\right ) \]
54675/32*x^8 + 375435/28*x^7 + 1597239/32*x^6 + 4750569/40*x^5 + 53086563/ 256*x^4 + 18842715/64*x^3 + 195497697/512*x^2 + 9077405/16*x - 99648703/10 24/(2*x - 1) + 389535839/1024*log(2*x - 1)
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.34 \[ \int \frac {(2+3 x)^7 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {1}{286720} \, {\left (2 \, x - 1\right )}^{8} {\left (\frac {45343800}{2 \, x - 1} + \frac {487438560}{{\left (2 \, x - 1\right )}^{2}} + \frac {3143702016}{{\left (2 \, x - 1\right )}^{3}} + \frac {13576070340}{{\left (2 \, x - 1\right )}^{4}} + \frac {41688082800}{{\left (2 \, x - 1\right )}^{5}} + \frac {96001584000}{{\left (2 \, x - 1\right )}^{6}} + \frac {189480773440}{{\left (2 \, x - 1\right )}^{7}} + 1913625\right )} - \frac {99648703}{1024 \, {\left (2 \, x - 1\right )}} - \frac {389535839}{1024} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) \]
1/286720*(2*x - 1)^8*(45343800/(2*x - 1) + 487438560/(2*x - 1)^2 + 3143702 016/(2*x - 1)^3 + 13576070340/(2*x - 1)^4 + 41688082800/(2*x - 1)^5 + 9600 1584000/(2*x - 1)^6 + 189480773440/(2*x - 1)^7 + 1913625) - 99648703/1024/ (2*x - 1) - 389535839/1024*log(1/2*abs(2*x - 1)/(2*x - 1)^2)
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^7 (3+5 x)^2}{(1-2 x)^2} \, dx=\frac {9077405\,x}{16}+\frac {389535839\,\ln \left (x-\frac {1}{2}\right )}{1024}-\frac {99648703}{2048\,\left (x-\frac {1}{2}\right )}+\frac {195497697\,x^2}{512}+\frac {18842715\,x^3}{64}+\frac {53086563\,x^4}{256}+\frac {4750569\,x^5}{40}+\frac {1597239\,x^6}{32}+\frac {375435\,x^7}{28}+\frac {54675\,x^8}{32} \]