Integrand size = 22, antiderivative size = 98 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^6} \, dx=\frac {484}{117649 (1-2 x)^2}+\frac {11264}{823543 (1-2 x)}-\frac {1}{1715 (2+3 x)^5}+\frac {16}{2401 (2+3 x)^4}-\frac {829}{50421 (2+3 x)^3}-\frac {2875}{117649 (2+3 x)^2}-\frac {24040}{823543 (2+3 x)}-\frac {11696 \log (1-2 x)}{823543}+\frac {11696 \log (2+3 x)}{823543} \]
484/117649/(1-2*x)^2+11264/823543/(1-2*x)-1/1715/(2+3*x)^5+16/2401/(2+3*x) ^4-829/50421/(2+3*x)^3-2875/117649/(2+3*x)^2-24040/823543/(2+3*x)-11696/82 3543*ln(1-2*x)+11696/823543*ln(2+3*x)
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^6} \, dx=\frac {4 \left (-\frac {7 \left (258089-4230956 x-19495039 x^2-14484765 x^3+36579240 x^4+63947880 x^5+28421280 x^6\right )}{4 (1-2 x)^2 (2+3 x)^5}-43860 \log (1-2 x)+43860 \log (4+6 x)\right )}{12353145} \]
(4*((-7*(258089 - 4230956*x - 19495039*x^2 - 14484765*x^3 + 36579240*x^4 + 63947880*x^5 + 28421280*x^6))/(4*(1 - 2*x)^2*(2 + 3*x)^5) - 43860*Log[1 - 2*x] + 43860*Log[4 + 6*x]))/12353145
Time = 0.23 (sec) , antiderivative size = 98, 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.
\(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x)^3 (3 x+2)^6} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {35088}{823543 (3 x+2)}+\frac {72120}{823543 (3 x+2)^2}+\frac {17250}{117649 (3 x+2)^3}+\frac {2487}{16807 (3 x+2)^4}-\frac {192}{2401 (3 x+2)^5}+\frac {3}{343 (3 x+2)^6}-\frac {23392}{823543 (2 x-1)}+\frac {22528}{823543 (2 x-1)^2}-\frac {1936}{117649 (2 x-1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {11264}{823543 (1-2 x)}-\frac {24040}{823543 (3 x+2)}+\frac {484}{117649 (1-2 x)^2}-\frac {2875}{117649 (3 x+2)^2}-\frac {829}{50421 (3 x+2)^3}+\frac {16}{2401 (3 x+2)^4}-\frac {1}{1715 (3 x+2)^5}-\frac {11696 \log (1-2 x)}{823543}+\frac {11696 \log (3 x+2)}{823543}\) |
484/(117649*(1 - 2*x)^2) + 11264/(823543*(1 - 2*x)) - 1/(1715*(2 + 3*x)^5) + 16/(2401*(2 + 3*x)^4) - 829/(50421*(2 + 3*x)^3) - 2875/(117649*(2 + 3*x )^2) - 24040/(823543*(2 + 3*x)) - (11696*Log[1 - 2*x])/823543 + (11696*Log [2 + 3*x])/823543
3.17.58.3.1 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 = 0.88 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.64
method | result | size |
norman | \(\frac {-\frac {4263192}{117649} x^{5}-\frac {2438616}{117649} x^{4}-\frac {1894752}{117649} x^{6}+\frac {965651}{117649} x^{3}+\frac {4230956}{1764735} x +\frac {19495039}{1764735} x^{2}-\frac {258089}{1764735}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{5}}-\frac {11696 \ln \left (-1+2 x \right )}{823543}+\frac {11696 \ln \left (2+3 x \right )}{823543}\) | \(63\) |
risch | \(\frac {-\frac {4263192}{117649} x^{5}-\frac {2438616}{117649} x^{4}-\frac {1894752}{117649} x^{6}+\frac {965651}{117649} x^{3}+\frac {4230956}{1764735} x +\frac {19495039}{1764735} x^{2}-\frac {258089}{1764735}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{5}}-\frac {11696 \ln \left (-1+2 x \right )}{823543}+\frac {11696 \ln \left (2+3 x \right )}{823543}\) | \(64\) |
default | \(\frac {484}{117649 \left (-1+2 x \right )^{2}}-\frac {11264}{823543 \left (-1+2 x \right )}-\frac {11696 \ln \left (-1+2 x \right )}{823543}-\frac {1}{1715 \left (2+3 x \right )^{5}}+\frac {16}{2401 \left (2+3 x \right )^{4}}-\frac {829}{50421 \left (2+3 x \right )^{3}}-\frac {2875}{117649 \left (2+3 x \right )^{2}}-\frac {24040}{823543 \left (2+3 x \right )}+\frac {11696 \ln \left (2+3 x \right )}{823543}\) | \(81\) |
parallelrisch | \(\frac {383358640 x -1571942400 \ln \left (\frac {2}{3}+x \right ) x^{3}-209592320 \ln \left (\frac {2}{3}+x \right ) x^{2}+209592320 \ln \left (\frac {2}{3}+x \right ) x -3978054297 x^{5}-756315252 x^{6}+585345852 x^{7}+575674680 x^{3}-3110640750 x^{4}+1388182320 x^{2}+1178956800 \ln \left (x -\frac {1}{2}\right ) x^{4}-1178956800 \ln \left (\frac {2}{3}+x \right ) x^{4}+59883520 \ln \left (\frac {2}{3}+x \right )+1571942400 \ln \left (x -\frac {1}{2}\right ) x^{3}+1818961920 \ln \left (\frac {2}{3}+x \right ) x^{7}+209592320 \ln \left (x -\frac {1}{2}\right ) x^{2}-209592320 \ln \left (x -\frac {1}{2}\right ) x +2475809280 \ln \left (\frac {2}{3}+x \right ) x^{5}+4244244480 \ln \left (\frac {2}{3}+x \right ) x^{6}-59883520 \ln \left (x -\frac {1}{2}\right )-1818961920 \ln \left (x -\frac {1}{2}\right ) x^{7}-4244244480 \ln \left (x -\frac {1}{2}\right ) x^{6}-2475809280 \ln \left (x -\frac {1}{2}\right ) x^{5}}{131766880 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{5}}\) | \(185\) |
(-4263192/117649*x^5-2438616/117649*x^4-1894752/117649*x^6+965651/117649*x ^3+4230956/1764735*x+19495039/1764735*x^2-258089/1764735)/(-1+2*x)^2/(2+3* x)^5-11696/823543*ln(-1+2*x)+11696/823543*ln(2+3*x)
Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.58 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^6} \, dx=-\frac {198948960 \, x^{6} + 447635160 \, x^{5} + 256054680 \, x^{4} - 101393355 \, x^{3} - 136465273 \, x^{2} - 175440 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 175440 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )} \log \left (2 \, x - 1\right ) - 29616692 \, x + 1806623}{12353145 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} \]
-1/12353145*(198948960*x^6 + 447635160*x^5 + 256054680*x^4 - 101393355*x^3 - 136465273*x^2 - 175440*(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x ^3 - 112*x^2 + 112*x + 32)*log(3*x + 2) + 175440*(972*x^7 + 2268*x^6 + 132 3*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)*log(2*x - 1) - 29616692* x + 1806623)/(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^6} \, dx=- \frac {28421280 x^{6} + 63947880 x^{5} + 36579240 x^{4} - 14484765 x^{3} - 19495039 x^{2} - 4230956 x + 258089}{1715322420 x^{7} + 4002418980 x^{6} + 2334744405 x^{5} - 1111783050 x^{4} - 1482377400 x^{3} - 197650320 x^{2} + 197650320 x + 56471520} - \frac {11696 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {11696 \log {\left (x + \frac {2}{3} \right )}}{823543} \]
-(28421280*x**6 + 63947880*x**5 + 36579240*x**4 - 14484765*x**3 - 19495039 *x**2 - 4230956*x + 258089)/(1715322420*x**7 + 4002418980*x**6 + 233474440 5*x**5 - 1111783050*x**4 - 1482377400*x**3 - 197650320*x**2 + 197650320*x + 56471520) - 11696*log(x - 1/2)/823543 + 11696*log(x + 2/3)/823543
Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^6} \, dx=-\frac {28421280 \, x^{6} + 63947880 \, x^{5} + 36579240 \, x^{4} - 14484765 \, x^{3} - 19495039 \, x^{2} - 4230956 \, x + 258089}{1764735 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} + \frac {11696}{823543} \, \log \left (3 \, x + 2\right ) - \frac {11696}{823543} \, \log \left (2 \, x - 1\right ) \]
-1/1764735*(28421280*x^6 + 63947880*x^5 + 36579240*x^4 - 14484765*x^3 - 19 495039*x^2 - 4230956*x + 258089)/(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32) + 11696/823543*log(3*x + 2) - 11696/8235 43*log(2*x - 1)
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.66 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^6} \, dx=-\frac {28421280 \, x^{6} + 63947880 \, x^{5} + 36579240 \, x^{4} - 14484765 \, x^{3} - 19495039 \, x^{2} - 4230956 \, x + 258089}{1764735 \, {\left (3 \, x + 2\right )}^{5} {\left (2 \, x - 1\right )}^{2}} + \frac {11696}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {11696}{823543} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-1/1764735*(28421280*x^6 + 63947880*x^5 + 36579240*x^4 - 14484765*x^3 - 19 495039*x^2 - 4230956*x + 258089)/((3*x + 2)^5*(2*x - 1)^2) + 11696/823543* log(abs(3*x + 2)) - 11696/823543*log(abs(2*x - 1))
Time = 1.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^6} \, dx=\frac {23392\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {5848\,x^6}{352947}+\frac {4386\,x^5}{117649}+\frac {203218\,x^4}{9529569}-\frac {965651\,x^3}{114354828}-\frac {19495039\,x^2}{1715322420}-\frac {1057739\,x}{428830605}+\frac {258089}{1715322420}}{x^7+\frac {7\,x^6}{3}+\frac {49\,x^5}{36}-\frac {35\,x^4}{54}-\frac {70\,x^3}{81}-\frac {28\,x^2}{243}+\frac {28\,x}{243}+\frac {8}{243}} \]
(23392*atanh((12*x)/7 + 1/7))/823543 - ((203218*x^4)/9529569 - (19495039*x ^2)/1715322420 - (965651*x^3)/114354828 - (1057739*x)/428830605 + (4386*x^ 5)/117649 + (5848*x^6)/352947 + 258089/1715322420)/((28*x)/243 - (28*x^2)/ 243 - (70*x^3)/81 - (35*x^4)/54 + (49*x^5)/36 + (7*x^6)/3 + x^7 + 8/243)