Integrand size = 20, antiderivative size = 69 \[ \int \frac {(2+3 x)^7 (3+5 x)}{(1-2 x)^2} \, dx=\frac {9058973}{512 (1-2 x)}+\frac {22333965 x}{256}+\frac {873207 x^2}{16}+\frac {2399985 x^3}{64}+\frac {1423899 x^4}{64}+\frac {793881 x^5}{80}+\frac {11421 x^6}{4}+\frac {10935 x^7}{28}+\frac {15647317}{256} \log (1-2 x) \]
9058973/512/(1-2*x)+22333965/256*x+873207/16*x^2+2399985/64*x^3+1423899/64 *x^4+793881/80*x^5+11421/4*x^6+10935/28*x^7+15647317/256*ln(1-2*x)
Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {(2+3 x)^7 (3+5 x)}{(1-2 x)^2} \, dx=\frac {1648903399-7692818118 x+4297526520 x^2+2567975760 x^3+1890599760 x^4+1239108192 x^5+608985216 x^6+190667520 x^7+27993600 x^8+2190624380 (-1+2 x) \log (1-2 x)}{35840 (-1+2 x)} \]
(1648903399 - 7692818118*x + 4297526520*x^2 + 2567975760*x^3 + 1890599760* x^4 + 1239108192*x^5 + 608985216*x^6 + 190667520*x^7 + 27993600*x^8 + 2190 624380*(-1 + 2*x)*Log[1 - 2*x])/(35840*(-1 + 2*x))
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.
| \(\displaystyle \int \frac {(3 x+2)^7 (5 x+3)}{(1-2 x)^2} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {10935 x^6}{4}+\frac {34263 x^5}{2}+\frac {793881 x^4}{16}+\frac {1423899 x^3}{16}+\frac {7199955 x^2}{64}+\frac {873207 x}{8}+\frac {15647317}{128 (2 x-1)}+\frac {9058973}{256 (2 x-1)^2}+\frac {22333965}{256}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {10935 x^7}{28}+\frac {11421 x^6}{4}+\frac {793881 x^5}{80}+\frac {1423899 x^4}{64}+\frac {2399985 x^3}{64}+\frac {873207 x^2}{16}+\frac {22333965 x}{256}+\frac {9058973}{512 (1-2 x)}+\frac {15647317}{256} \log (1-2 x)\) |
9058973/(512*(1 - 2*x)) + (22333965*x)/256 + (873207*x^2)/16 + (2399985*x^ 3)/64 + (1423899*x^4)/64 + (793881*x^5)/80 + (11421*x^6)/4 + (10935*x^7)/2 8 + (15647317*Log[1 - 2*x])/256
3.16.39.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Time = 0.85 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {10935 x^{7}}{28}+\frac {11421 x^{6}}{4}+\frac {793881 x^{5}}{80}+\frac {1423899 x^{4}}{64}+\frac {2399985 x^{3}}{64}+\frac {873207 x^{2}}{16}+\frac {22333965 x}{256}-\frac {9058973}{1024 \left (x -\frac {1}{2}\right )}+\frac {15647317 \ln \left (-1+2 x \right )}{256}\) | \(50\) |
| default | \(\frac {10935 x^{7}}{28}+\frac {11421 x^{6}}{4}+\frac {793881 x^{5}}{80}+\frac {1423899 x^{4}}{64}+\frac {2399985 x^{3}}{64}+\frac {873207 x^{2}}{16}+\frac {22333965 x}{256}+\frac {15647317 \ln \left (-1+2 x \right )}{256}-\frac {9058973}{512 \left (-1+2 x \right )}\) | \(52\) |
| norman | \(\frac {-\frac {15696469}{128} x +\frac {15348309}{128} x^{2}+\frac {4585671}{64} x^{3}+\frac {3376071}{64} x^{4}+\frac {5531733}{160} x^{5}+\frac {679671}{40} x^{6}+\frac {148959}{28} x^{7}+\frac {10935}{14} x^{8}}{-1+2 x}+\frac {15647317 \ln \left (-1+2 x \right )}{256}\) | \(57\) |
| parallelrisch | \(\frac {6998400 x^{8}+47666880 x^{7}+152246304 x^{6}+309777048 x^{5}+472649940 x^{4}+641993940 x^{3}+1095312190 \ln \left (x -\frac {1}{2}\right ) x +1074381630 x^{2}-547656095 \ln \left (x -\frac {1}{2}\right )-1098752830 x}{-8960+17920 x}\) | \(62\) |
| meijerg | \(\frac {2720 x}{1-2 x}+\frac {15647317 \ln \left (1-2 x \right )}{256}+\frac {2072 x \left (-6 x +6\right )}{1-2 x}+\frac {4725 x \left (-8 x^{2}-12 x +12\right )}{2 \left (1-2 x \right )}+\frac {1197 x \left (-40 x^{3}-40 x^{2}-60 x +60\right )}{2 \left (1-2 x \right )}+\frac {14553 x \left (-48 x^{4}-40 x^{3}-40 x^{2}-60 x +60\right )}{32 \left (1-2 x \right )}+\frac {9477 x \left (-448 x^{5}-336 x^{4}-280 x^{3}-280 x^{2}-420 x +420\right )}{320 \left (1-2 x \right )}+\frac {19197 x \left (-1280 x^{6}-896 x^{5}-672 x^{4}-560 x^{3}-560 x^{2}-840 x +840\right )}{5120 \left (1-2 x \right )}+\frac {243 x \left (-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{1792 \left (1-2 x \right )}\) | \(230\) |
10935/28*x^7+11421/4*x^6+793881/80*x^5+1423899/64*x^4+2399985/64*x^3+87320 7/16*x^2+22333965/256*x-9058973/1024/(x-1/2)+15647317/256*ln(-1+2*x)
Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^7 (3+5 x)}{(1-2 x)^2} \, dx=\frac {13996800 \, x^{8} + 95333760 \, x^{7} + 304492608 \, x^{6} + 619554096 \, x^{5} + 945299880 \, x^{4} + 1283987880 \, x^{3} + 2148763260 \, x^{2} + 1095312190 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 1563377550 \, x - 317064055}{17920 \, {\left (2 \, x - 1\right )}} \]
1/17920*(13996800*x^8 + 95333760*x^7 + 304492608*x^6 + 619554096*x^5 + 945 299880*x^4 + 1283987880*x^3 + 2148763260*x^2 + 1095312190*(2*x - 1)*log(2* x - 1) - 1563377550*x - 317064055)/(2*x - 1)
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^7 (3+5 x)}{(1-2 x)^2} \, dx=\frac {10935 x^{7}}{28} + \frac {11421 x^{6}}{4} + \frac {793881 x^{5}}{80} + \frac {1423899 x^{4}}{64} + \frac {2399985 x^{3}}{64} + \frac {873207 x^{2}}{16} + \frac {22333965 x}{256} + \frac {15647317 \log {\left (2 x - 1 \right )}}{256} - \frac {9058973}{1024 x - 512} \]
10935*x**7/28 + 11421*x**6/4 + 793881*x**5/80 + 1423899*x**4/64 + 2399985* x**3/64 + 873207*x**2/16 + 22333965*x/256 + 15647317*log(2*x - 1)/256 - 90 58973/(1024*x - 512)
Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^7 (3+5 x)}{(1-2 x)^2} \, dx=\frac {10935}{28} \, x^{7} + \frac {11421}{4} \, x^{6} + \frac {793881}{80} \, x^{5} + \frac {1423899}{64} \, x^{4} + \frac {2399985}{64} \, x^{3} + \frac {873207}{16} \, x^{2} + \frac {22333965}{256} \, x - \frac {9058973}{512 \, {\left (2 \, x - 1\right )}} + \frac {15647317}{256} \, \log \left (2 \, x - 1\right ) \]
10935/28*x^7 + 11421/4*x^6 + 793881/80*x^5 + 1423899/64*x^4 + 2399985/64*x ^3 + 873207/16*x^2 + 22333965/256*x - 9058973/512/(2*x - 1) + 15647317/256 *log(2*x - 1)
Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.35 \[ \int \frac {(2+3 x)^7 (3+5 x)}{(1-2 x)^2} \, dx=\frac {3}{35840} \, {\left (2 \, x - 1\right )}^{7} {\left (\frac {788130}{2 \, x - 1} + \frac {7668108}{{\left (2 \, x - 1\right )}^{2}} + \frac {44406495}{{\left (2 \, x - 1\right )}^{3}} + \frac {171431400}{{\left (2 \, x - 1\right )}^{4}} + \frac {476478450}{{\left (2 \, x - 1\right )}^{5}} + \frac {1103547620}{{\left (2 \, x - 1\right )}^{6}} + 36450\right )} - \frac {9058973}{512 \, {\left (2 \, x - 1\right )}} - \frac {15647317}{256} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) \]
3/35840*(2*x - 1)^7*(788130/(2*x - 1) + 7668108/(2*x - 1)^2 + 44406495/(2* x - 1)^3 + 171431400/(2*x - 1)^4 + 476478450/(2*x - 1)^5 + 1103547620/(2*x - 1)^6 + 36450) - 9058973/512/(2*x - 1) - 15647317/256*log(1/2*abs(2*x - 1)/(2*x - 1)^2)
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^7 (3+5 x)}{(1-2 x)^2} \, dx=\frac {22333965\,x}{256}+\frac {15647317\,\ln \left (x-\frac {1}{2}\right )}{256}-\frac {9058973}{1024\,\left (x-\frac {1}{2}\right )}+\frac {873207\,x^2}{16}+\frac {2399985\,x^3}{64}+\frac {1423899\,x^4}{64}+\frac {793881\,x^5}{80}+\frac {11421\,x^6}{4}+\frac {10935\,x^7}{28} \]