Integrand size = 22, antiderivative size = 109 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^8} \, dx=\frac {3872}{5764801 (1-2 x)}-\frac {1}{1029 (2+3 x)^7}+\frac {11}{1029 (2+3 x)^6}-\frac {319}{12005 (2+3 x)^5}-\frac {341}{16807 (2+3 x)^4}-\frac {4180}{352947 (2+3 x)^3}-\frac {5632}{823543 (2+3 x)^2}-\frac {4048}{823543 (2+3 x)}-\frac {68288 \log (1-2 x)}{40353607}+\frac {68288 \log (2+3 x)}{40353607} \]
3872/5764801/(1-2*x)-1/1029/(2+3*x)^7+11/1029/(2+3*x)^6-319/12005/(2+3*x)^ 5-341/16807/(2+3*x)^4-4180/352947/(2+3*x)^3-5632/823543/(2+3*x)^2-4048/823 543/(2+3*x)-68288/40353607*ln(1-2*x)+68288/40353607*ln(2+3*x)
Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.68 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^8} \, dx=\frac {16 \left (-\frac {7 \left (-76539293-327016403 x-183177225 x^2+1495734471 x^3+4176440730 x^4+5057708040 x^5+3049144560 x^6+746729280 x^7\right )}{16 (-1+2 x) (2+3 x)^7}-64020 \log (1-2 x)+64020 \log (4+6 x)\right )}{605304105} \]
(16*((-7*(-76539293 - 327016403*x - 183177225*x^2 + 1495734471*x^3 + 41764 40730*x^4 + 5057708040*x^5 + 3049144560*x^6 + 746729280*x^7))/(16*(-1 + 2* x)*(2 + 3*x)^7) - 64020*Log[1 - 2*x] + 64020*Log[4 + 6*x]))/605304105
Time = 0.24 (sec) , antiderivative size = 109, 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.
| \(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x)^2 (3 x+2)^8} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {204864}{40353607 (3 x+2)}+\frac {12144}{823543 (3 x+2)^2}+\frac {33792}{823543 (3 x+2)^3}+\frac {12540}{117649 (3 x+2)^4}+\frac {4092}{16807 (3 x+2)^5}+\frac {957}{2401 (3 x+2)^6}-\frac {66}{343 (3 x+2)^7}+\frac {1}{49 (3 x+2)^8}-\frac {136576}{40353607 (2 x-1)}+\frac {7744}{5764801 (2 x-1)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3872}{5764801 (1-2 x)}-\frac {4048}{823543 (3 x+2)}-\frac {5632}{823543 (3 x+2)^2}-\frac {4180}{352947 (3 x+2)^3}-\frac {341}{16807 (3 x+2)^4}-\frac {319}{12005 (3 x+2)^5}+\frac {11}{1029 (3 x+2)^6}-\frac {1}{1029 (3 x+2)^7}-\frac {68288 \log (1-2 x)}{40353607}+\frac {68288 \log (3 x+2)}{40353607}\) |
3872/(5764801*(1 - 2*x)) - 1/(1029*(2 + 3*x)^7) + 11/(1029*(2 + 3*x)^6) - 319/(12005*(2 + 3*x)^5) - 341/(16807*(2 + 3*x)^4) - 4180/(352947*(2 + 3*x) ^3) - 5632/(823543*(2 + 3*x)^2) - 4048/(823543*(2 + 3*x)) - (68288*Log[1 - 2*x])/40353607 + (68288*Log[2 + 3*x])/40353607
3.16.70.3.1 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 = 2.65 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62
| method | result | size |
| norman | \(\frac {-\frac {71225451}{4117715} x^{3}-\frac {49781952}{5764801} x^{7}-\frac {48168648}{823543} x^{5}-\frac {39775626}{823543} x^{4}-\frac {4148496}{117649} x^{6}+\frac {1744545}{823543} x^{2}+\frac {46716629}{12353145} x +\frac {76539293}{86472015}}{\left (-1+2 x \right ) \left (2+3 x \right )^{7}}-\frac {68288 \ln \left (-1+2 x \right )}{40353607}+\frac {68288 \ln \left (2+3 x \right )}{40353607}\) | \(68\) |
| risch | \(\frac {-\frac {71225451}{4117715} x^{3}-\frac {49781952}{5764801} x^{7}-\frac {48168648}{823543} x^{5}-\frac {39775626}{823543} x^{4}-\frac {4148496}{117649} x^{6}+\frac {1744545}{823543} x^{2}+\frac {46716629}{12353145} x +\frac {76539293}{86472015}}{\left (-1+2 x \right ) \left (2+3 x \right )^{7}}-\frac {68288 \ln \left (-1+2 x \right )}{40353607}+\frac {68288 \ln \left (2+3 x \right )}{40353607}\) | \(69\) |
| default | \(-\frac {3872}{5764801 \left (-1+2 x \right )}-\frac {68288 \ln \left (-1+2 x \right )}{40353607}-\frac {1}{1029 \left (2+3 x \right )^{7}}+\frac {11}{1029 \left (2+3 x \right )^{6}}-\frac {319}{12005 \left (2+3 x \right )^{5}}-\frac {341}{16807 \left (2+3 x \right )^{4}}-\frac {4180}{352947 \left (2+3 x \right )^{3}}-\frac {5632}{823543 \left (2+3 x \right )^{2}}-\frac {4048}{823543 \left (2+3 x \right )}+\frac {68288 \ln \left (2+3 x \right )}{40353607}\) | \(90\) |
| parallelrisch | \(\frac {-96638852800 x -132161863680 \ln \left (\frac {2}{3}+x \right ) x^{3}-146846515200 \ln \left (\frac {2}{3}+x \right ) x^{2}-47550300160 \ln \left (\frac {2}{3}+x \right ) x +2944936522836 x^{5}+4557442328586 x^{6}+3031810289865 x^{7}-986787280080 x^{3}+102789497160 x^{4}-545359125920 x^{2}+781160024358 x^{8}-330404659200 \ln \left (x -\frac {1}{2}\right ) x^{4}+330404659200 \ln \left (\frac {2}{3}+x \right ) x^{4}-5594152960 \ln \left (\frac {2}{3}+x \right )+132161863680 \ln \left (x -\frac {1}{2}\right ) x^{3}+796511232000 \ln \left (\frac {2}{3}+x \right ) x^{7}+146846515200 \ln \left (x -\frac {1}{2}\right ) x^{2}+47550300160 \ln \left (x -\frac {1}{2}\right ) x +1090335375360 \ln \left (\frac {2}{3}+x \right ) x^{5}+1338138869760 \ln \left (\frac {2}{3}+x \right ) x^{6}-191162695680 \ln \left (x -\frac {1}{2}\right ) x^{8}+5594152960 \ln \left (x -\frac {1}{2}\right )+191162695680 \ln \left (\frac {2}{3}+x \right ) x^{8}-796511232000 \ln \left (x -\frac {1}{2}\right ) x^{7}-1338138869760 \ln \left (x -\frac {1}{2}\right ) x^{6}-1090335375360 \ln \left (x -\frac {1}{2}\right ) x^{5}}{25826308480 \left (-1+2 x \right ) \left (2+3 x \right )^{7}}\) | \(208\) |
(-71225451/4117715*x^3-49781952/5764801*x^7-48168648/823543*x^5-39775626/8 23543*x^4-4148496/117649*x^6+1744545/823543*x^2+46716629/12353145*x+765392 93/86472015)/(-1+2*x)/(2+3*x)^7-68288/40353607*ln(-1+2*x)+68288/40353607*l n(2+3*x)
Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.61 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^8} \, dx=-\frac {5227104960 \, x^{7} + 21344011920 \, x^{6} + 35403956280 \, x^{5} + 29235085110 \, x^{4} + 10470141297 \, x^{3} - 1282240575 \, x^{2} - 1024320 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )} \log \left (3 \, x + 2\right ) + 1024320 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )} \log \left (2 \, x - 1\right ) - 2289114821 \, x - 535775051}{605304105 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )}} \]
-1/605304105*(5227104960*x^7 + 21344011920*x^6 + 35403956280*x^5 + 2923508 5110*x^4 + 10470141297*x^3 - 1282240575*x^2 - 1024320*(4374*x^8 + 18225*x^ 7 + 30618*x^6 + 24948*x^5 + 7560*x^4 - 3024*x^3 - 3360*x^2 - 1088*x - 128) *log(3*x + 2) + 1024320*(4374*x^8 + 18225*x^7 + 30618*x^6 + 24948*x^5 + 75 60*x^4 - 3024*x^3 - 3360*x^2 - 1088*x - 128)*log(2*x - 1) - 2289114821*x - 535775051)/(4374*x^8 + 18225*x^7 + 30618*x^6 + 24948*x^5 + 7560*x^4 - 302 4*x^3 - 3360*x^2 - 1088*x - 128)
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^8} \, dx=\frac {- 746729280 x^{7} - 3049144560 x^{6} - 5057708040 x^{5} - 4176440730 x^{4} - 1495734471 x^{3} + 183177225 x^{2} + 327016403 x + 76539293}{378228593610 x^{8} + 1575952473375 x^{7} + 2647600155270 x^{6} + 2157303830220 x^{5} + 653728433400 x^{4} - 261491373360 x^{3} - 290545970400 x^{2} - 94081552320 x - 11068417920} - \frac {68288 \log {\left (x - \frac {1}{2} \right )}}{40353607} + \frac {68288 \log {\left (x + \frac {2}{3} \right )}}{40353607} \]
(-746729280*x**7 - 3049144560*x**6 - 5057708040*x**5 - 4176440730*x**4 - 1 495734471*x**3 + 183177225*x**2 + 327016403*x + 76539293)/(378228593610*x* *8 + 1575952473375*x**7 + 2647600155270*x**6 + 2157303830220*x**5 + 653728 433400*x**4 - 261491373360*x**3 - 290545970400*x**2 - 94081552320*x - 1106 8417920) - 68288*log(x - 1/2)/40353607 + 68288*log(x + 2/3)/40353607
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^8} \, dx=-\frac {746729280 \, x^{7} + 3049144560 \, x^{6} + 5057708040 \, x^{5} + 4176440730 \, x^{4} + 1495734471 \, x^{3} - 183177225 \, x^{2} - 327016403 \, x - 76539293}{86472015 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )}} + \frac {68288}{40353607} \, \log \left (3 \, x + 2\right ) - \frac {68288}{40353607} \, \log \left (2 \, x - 1\right ) \]
-1/86472015*(746729280*x^7 + 3049144560*x^6 + 5057708040*x^5 + 4176440730* x^4 + 1495734471*x^3 - 183177225*x^2 - 327016403*x - 76539293)/(4374*x^8 + 18225*x^7 + 30618*x^6 + 24948*x^5 + 7560*x^4 - 3024*x^3 - 3360*x^2 - 1088 *x - 128) + 68288/40353607*log(3*x + 2) - 68288/40353607*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^8} \, dx=-\frac {3872}{5764801 \, {\left (2 \, x - 1\right )}} + \frac {16 \, {\left (\frac {6995041011}{2 \, x - 1} + \frac {43950177747}{{\left (2 \, x - 1\right )}^{2}} + \frac {148454802405}{{\left (2 \, x - 1\right )}^{3}} + \frac {284722344900}{{\left (2 \, x - 1\right )}^{4}} + \frac {294251913900}{{\left (2 \, x - 1\right )}^{5}} + \frac {128036230210}{{\left (2 \, x - 1\right )}^{6}} + 466999587\right )}}{1412376245 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{7}} + \frac {68288}{40353607} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]
-3872/5764801/(2*x - 1) + 16/1412376245*(6995041011/(2*x - 1) + 4395017774 7/(2*x - 1)^2 + 148454802405/(2*x - 1)^3 + 284722344900/(2*x - 1)^4 + 2942 51913900/(2*x - 1)^5 + 128036230210/(2*x - 1)^6 + 466999587)/(7/(2*x - 1) + 3)^7 + 68288/40353607*log(abs(-7/(2*x - 1) - 3))
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.79 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^8} \, dx=\frac {136576\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{40353607}-\frac {\frac {34144\,x^7}{17294403}+\frac {8536\,x^6}{1058841}+\frac {892012\,x^5}{66706983}+\frac {2209757\,x^4}{200120949}+\frac {7913939\,x^3}{2001209490}-\frac {581515\,x^2}{1200725694}-\frac {46716629\,x}{54032656230}-\frac {76539293}{378228593610}}{x^8+\frac {25\,x^7}{6}+7\,x^6+\frac {154\,x^5}{27}+\frac {140\,x^4}{81}-\frac {56\,x^3}{81}-\frac {560\,x^2}{729}-\frac {544\,x}{2187}-\frac {64}{2187}} \]
(136576*atanh((12*x)/7 + 1/7))/40353607 - ((7913939*x^3)/2001209490 - (581 515*x^2)/1200725694 - (46716629*x)/54032656230 + (2209757*x^4)/200120949 + (892012*x^5)/66706983 + (8536*x^6)/1058841 + (34144*x^7)/17294403 - 76539 293/378228593610)/((140*x^4)/81 - (560*x^2)/729 - (56*x^3)/81 - (544*x)/21 87 + (154*x^5)/27 + 7*x^6 + (25*x^7)/6 + x^8 - 64/2187)