Integrand size = 22, antiderivative size = 87 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^6} \, dx=\frac {968}{117649 (1-2 x)}-\frac {1}{735 (2+3 x)^5}+\frac {11}{686 (2+3 x)^4}-\frac {319}{7203 (2+3 x)^3}-\frac {682}{16807 (2+3 x)^2}-\frac {4180}{117649 (2+3 x)}-\frac {11264 \log (1-2 x)}{823543}+\frac {11264 \log (2+3 x)}{823543} \]
968/117649/(1-2*x)-1/735/(2+3*x)^5+11/686/(2+3*x)^4-319/7203/(2+3*x)^3-682 /16807/(2+3*x)^2-4180/117649/(2+3*x)-11264/823543*ln(1-2*x)+11264/823543*l n(2+3*x)
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^6} \, dx=\frac {8 \left (-\frac {21 \left (-913244-1530877 x+7494080 x^2+24288000 x^3+25090560 x^4+9123840 x^5\right )}{16 (-1+2 x) (2+3 x)^5}-21120 \log (1-2 x)+21120 \log (4+6 x)\right )}{12353145} \]
(8*((-21*(-913244 - 1530877*x + 7494080*x^2 + 24288000*x^3 + 25090560*x^4 + 9123840*x^5))/(16*(-1 + 2*x)*(2 + 3*x)^5) - 21120*Log[1 - 2*x] + 21120*L og[4 + 6*x]))/12353145
Time = 0.21 (sec) , antiderivative size = 87, 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.
\(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x)^2 (3 x+2)^6} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {33792}{823543 (3 x+2)}+\frac {12540}{117649 (3 x+2)^2}+\frac {4092}{16807 (3 x+2)^3}+\frac {957}{2401 (3 x+2)^4}-\frac {66}{343 (3 x+2)^5}+\frac {1}{49 (3 x+2)^6}-\frac {22528}{823543 (2 x-1)}+\frac {1936}{117649 (2 x-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {968}{117649 (1-2 x)}-\frac {4180}{117649 (3 x+2)}-\frac {682}{16807 (3 x+2)^2}-\frac {319}{7203 (3 x+2)^3}+\frac {11}{686 (3 x+2)^4}-\frac {1}{735 (3 x+2)^5}-\frac {11264 \log (1-2 x)}{823543}+\frac {11264 \log (3 x+2)}{823543}\) |
968/(117649*(1 - 2*x)) - 1/(735*(2 + 3*x)^5) + 11/(686*(2 + 3*x)^4) - 319/ (7203*(2 + 3*x)^3) - 682/(16807*(2 + 3*x)^2) - 4180/(117649*(2 + 3*x)) - ( 11264*Log[1 - 2*x])/823543 + (11264*Log[2 + 3*x])/823543
3.16.68.3.1 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 = 58, normalized size of antiderivative = 0.67
method | result | size |
norman | \(\frac {-\frac {2509056}{117649} x^{4}-\frac {2428800}{117649} x^{3}-\frac {912384}{117649} x^{5}-\frac {749408}{117649} x^{2}+\frac {1530877}{1176490} x +\frac {456622}{588245}}{\left (-1+2 x \right ) \left (2+3 x \right )^{5}}-\frac {11264 \ln \left (-1+2 x \right )}{823543}+\frac {11264 \ln \left (2+3 x \right )}{823543}\) | \(58\) |
risch | \(\frac {-\frac {2509056}{117649} x^{4}-\frac {2428800}{117649} x^{3}-\frac {912384}{117649} x^{5}-\frac {749408}{117649} x^{2}+\frac {1530877}{1176490} x +\frac {456622}{588245}}{\left (-1+2 x \right ) \left (2+3 x \right )^{5}}-\frac {11264 \ln \left (-1+2 x \right )}{823543}+\frac {11264 \ln \left (2+3 x \right )}{823543}\) | \(59\) |
default | \(-\frac {968}{117649 \left (-1+2 x \right )}-\frac {11264 \ln \left (-1+2 x \right )}{823543}-\frac {1}{735 \left (2+3 x \right )^{5}}+\frac {11}{686 \left (2+3 x \right )^{4}}-\frac {319}{7203 \left (2+3 x \right )^{3}}-\frac {682}{16807 \left (2+3 x \right )^{2}}-\frac {4180}{117649 \left (2+3 x \right )}+\frac {11264 \ln \left (2+3 x \right )}{823543}\) | \(72\) |
parallelrisch | \(\frac {-391100080 x +648806400 \ln \left (\frac {2}{3}+x \right ) x^{3}-432537600 \ln \left (\frac {2}{3}+x \right ) x^{2}-317194240 \ln \left (\frac {2}{3}+x \right ) x +3379509378 x^{5}+1553428044 x^{6}-1569568560 x^{3}+1504935180 x^{4}-1606461920 x^{2}-2433024000 \ln \left (x -\frac {1}{2}\right ) x^{4}+2433024000 \ln \left (\frac {2}{3}+x \right ) x^{4}-57671680 \ln \left (\frac {2}{3}+x \right )-648806400 \ln \left (x -\frac {1}{2}\right ) x^{3}+432537600 \ln \left (x -\frac {1}{2}\right ) x^{2}+317194240 \ln \left (x -\frac {1}{2}\right ) x +2481684480 \ln \left (\frac {2}{3}+x \right ) x^{5}+875888640 \ln \left (\frac {2}{3}+x \right ) x^{6}+57671680 \ln \left (x -\frac {1}{2}\right )-875888640 \ln \left (x -\frac {1}{2}\right ) x^{6}-2481684480 \ln \left (x -\frac {1}{2}\right ) x^{5}}{131766880 \left (-1+2 x \right ) \left (2+3 x \right )^{5}}\) | \(162\) |
(-2509056/117649*x^4-2428800/117649*x^3-912384/117649*x^5-749408/117649*x^ 2+1530877/1176490*x+456622/588245)/(-1+2*x)/(2+3*x)^5-11264/823543*ln(-1+2 *x)+11264/823543*ln(2+3*x)
Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.55 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^6} \, dx=-\frac {63866880 \, x^{5} + 175633920 \, x^{4} + 170016000 \, x^{3} + 52458560 \, x^{2} - 112640 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (3 \, x + 2\right ) + 112640 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (2 \, x - 1\right ) - 10716139 \, x - 6392708}{8235430 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \]
-1/8235430*(63866880*x^5 + 175633920*x^4 + 170016000*x^3 + 52458560*x^2 - 112640*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*lo g(3*x + 2) + 112640*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 1 76*x - 32)*log(2*x - 1) - 10716139*x - 6392708)/(486*x^6 + 1377*x^5 + 1350 *x^4 + 360*x^3 - 240*x^2 - 176*x - 32)
Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^6} \, dx=\frac {- 9123840 x^{5} - 25090560 x^{4} - 24288000 x^{3} - 7494080 x^{2} + 1530877 x + 913244}{571774140 x^{6} + 1620026730 x^{5} + 1588261500 x^{4} + 423536400 x^{3} - 282357600 x^{2} - 207062240 x - 37647680} - \frac {11264 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {11264 \log {\left (x + \frac {2}{3} \right )}}{823543} \]
(-9123840*x**5 - 25090560*x**4 - 24288000*x**3 - 7494080*x**2 + 1530877*x + 913244)/(571774140*x**6 + 1620026730*x**5 + 1588261500*x**4 + 423536400* x**3 - 282357600*x**2 - 207062240*x - 37647680) - 11264*log(x - 1/2)/82354 3 + 11264*log(x + 2/3)/823543
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^6} \, dx=-\frac {9123840 \, x^{5} + 25090560 \, x^{4} + 24288000 \, x^{3} + 7494080 \, x^{2} - 1530877 \, x - 913244}{1176490 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} + \frac {11264}{823543} \, \log \left (3 \, x + 2\right ) - \frac {11264}{823543} \, \log \left (2 \, x - 1\right ) \]
-1/1176490*(9123840*x^5 + 25090560*x^4 + 24288000*x^3 + 7494080*x^2 - 1530 877*x - 913244)/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32) + 11264/823543*log(3*x + 2) - 11264/823543*log(2*x - 1)
Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^6} \, dx=-\frac {968}{117649 \, {\left (2 \, x - 1\right )}} + \frac {8 \, {\left (\frac {18039105}{2 \, x - 1} + \frac {68101425}{{\left (2 \, x - 1\right )}^{2}} + \frac {114476250}{{\left (2 \, x - 1\right )}^{3}} + \frac {72150050}{{\left (2 \, x - 1\right )}^{4}} + 1800144\right )}}{4117715 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{5}} + \frac {11264}{823543} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]
-968/117649/(2*x - 1) + 8/4117715*(18039105/(2*x - 1) + 68101425/(2*x - 1) ^2 + 114476250/(2*x - 1)^3 + 72150050/(2*x - 1)^4 + 1800144)/(7/(2*x - 1) + 3)^5 + 11264/823543*log(abs(-7/(2*x - 1) - 3))
Time = 1.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^6} \, dx=\frac {22528\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {5632\,x^5}{352947}+\frac {15488\,x^4}{352947}+\frac {404800\,x^3}{9529569}+\frac {374704\,x^2}{28588707}-\frac {1530877\,x}{571774140}-\frac {228311}{142943535}}{x^6+\frac {17\,x^5}{6}+\frac {25\,x^4}{9}+\frac {20\,x^3}{27}-\frac {40\,x^2}{81}-\frac {88\,x}{243}-\frac {16}{243}} \]