Integrand size = 22, antiderivative size = 87 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^5} \, dx=\frac {242}{16807 (1-2 x)^2}+\frac {4180}{117649 (1-2 x)}-\frac {1}{1372 (2+3 x)^4}+\frac {64}{7203 (2+3 x)^3}-\frac {829}{33614 (2+3 x)^2}-\frac {5750}{117649 (2+3 x)}-\frac {24040 \log (1-2 x)}{823543}+\frac {24040 \log (2+3 x)}{823543} \]
242/16807/(1-2*x)^2+4180/117649/(1-2*x)-1/1372/(2+3*x)^4+64/7203/(2+3*x)^3 -829/33614/(2+3*x)^2-5750/117649/(2+3*x)-24040/823543*ln(1-2*x)+24040/8235 43*ln(2+3*x)
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^5} \, dx=\frac {2 \left (-\frac {7 \left (-460595-4966396 x-10343210 x^2+3606000 x^3+24665040 x^4+15577920 x^5\right )}{8 (1-2 x)^2 (2+3 x)^4}-36060 \log (1-2 x)+36060 \log (4+6 x)\right )}{2470629} \]
(2*((-7*(-460595 - 4966396*x - 10343210*x^2 + 3606000*x^3 + 24665040*x^4 + 15577920*x^5))/(8*(1 - 2*x)^2*(2 + 3*x)^4) - 36060*Log[1 - 2*x] + 36060*L og[4 + 6*x]))/2470629
Time = 0.22 (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.
\(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x)^3 (3 x+2)^5} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {72120}{823543 (3 x+2)}+\frac {17250}{117649 (3 x+2)^2}+\frac {2487}{16807 (3 x+2)^3}-\frac {192}{2401 (3 x+2)^4}+\frac {3}{343 (3 x+2)^5}-\frac {48080}{823543 (2 x-1)}+\frac {8360}{117649 (2 x-1)^2}-\frac {968}{16807 (2 x-1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4180}{117649 (1-2 x)}-\frac {5750}{117649 (3 x+2)}+\frac {242}{16807 (1-2 x)^2}-\frac {829}{33614 (3 x+2)^2}+\frac {64}{7203 (3 x+2)^3}-\frac {1}{1372 (3 x+2)^4}-\frac {24040 \log (1-2 x)}{823543}+\frac {24040 \log (3 x+2)}{823543}\) |
242/(16807*(1 - 2*x)^2) + 4180/(117649*(1 - 2*x)) - 1/(1372*(2 + 3*x)^4) + 64/(7203*(2 + 3*x)^3) - 829/(33614*(2 + 3*x)^2) - 5750/(117649*(2 + 3*x)) - (24040*Log[1 - 2*x])/823543 + (24040*Log[2 + 3*x])/823543
3.17.57.3.1 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 = 58, normalized size of antiderivative = 0.67
method | result | size |
norman | \(\frac {-\frac {2055420}{117649} x^{4}-\frac {1298160}{117649} x^{5}-\frac {300500}{117649} x^{3}+\frac {1241599}{352947} x +\frac {5171605}{705894} x^{2}+\frac {460595}{1411788}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{4}}-\frac {24040 \ln \left (-1+2 x \right )}{823543}+\frac {24040 \ln \left (2+3 x \right )}{823543}\) | \(58\) |
risch | \(\frac {-\frac {2055420}{117649} x^{4}-\frac {1298160}{117649} x^{5}-\frac {300500}{117649} x^{3}+\frac {1241599}{352947} x +\frac {5171605}{705894} x^{2}+\frac {460595}{1411788}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{4}}-\frac {24040 \ln \left (-1+2 x \right )}{823543}+\frac {24040 \ln \left (2+3 x \right )}{823543}\) | \(59\) |
default | \(\frac {242}{16807 \left (-1+2 x \right )^{2}}-\frac {4180}{117649 \left (-1+2 x \right )}-\frac {24040 \ln \left (-1+2 x \right )}{823543}-\frac {1}{1372 \left (2+3 x \right )^{4}}+\frac {64}{7203 \left (2+3 x \right )^{3}}-\frac {829}{33614 \left (2+3 x \right )^{2}}-\frac {5750}{117649 \left (2+3 x \right )}+\frac {24040 \ln \left (2+3 x \right )}{823543}\) | \(72\) |
parallelrisch | \(\frac {151021024 x -406179840 \ln \left (\frac {2}{3}+x \right ) x^{3}-160010240 \ln \left (\frac {2}{3}+x \right ) x^{2}+49233920 \ln \left (\frac {2}{3}+x \right ) x -1161925380 x^{5}-348209820 x^{6}+149102520 x^{3}-1007880615 x^{4}+497917560 x^{2}-124623360 \ln \left (x -\frac {1}{2}\right ) x^{4}+124623360 \ln \left (\frac {2}{3}+x \right ) x^{4}+24616960 \ln \left (\frac {2}{3}+x \right )+406179840 \ln \left (x -\frac {1}{2}\right ) x^{3}+160010240 \ln \left (x -\frac {1}{2}\right ) x^{2}-49233920 \ln \left (x -\frac {1}{2}\right ) x +830822400 \ln \left (\frac {2}{3}+x \right ) x^{5}+498493440 \ln \left (\frac {2}{3}+x \right ) x^{6}-24616960 \ln \left (x -\frac {1}{2}\right )-498493440 \ln \left (x -\frac {1}{2}\right ) x^{6}-830822400 \ln \left (x -\frac {1}{2}\right ) x^{5}}{52706752 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{4}}\) | \(162\) |
(-2055420/117649*x^4-1298160/117649*x^5-300500/117649*x^3+1241599/352947*x +5171605/705894*x^2+460595/1411788)/(-1+2*x)^2/(2+3*x)^4-24040/823543*ln(- 1+2*x)+24040/823543*ln(2+3*x)
Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.55 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^5} \, dx=-\frac {109045440 \, x^{5} + 172655280 \, x^{4} + 25242000 \, x^{3} - 72402470 \, x^{2} - 288480 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 288480 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (2 \, x - 1\right ) - 34764772 \, x - 3224165}{9882516 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]
-1/9882516*(109045440*x^5 + 172655280*x^4 + 25242000*x^3 - 72402470*x^2 - 288480*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*log(3* x + 2) + 288480*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 1 6)*log(2*x - 1) - 34764772*x - 3224165)/(324*x^6 + 540*x^5 + 81*x^4 - 264* x^3 - 104*x^2 + 32*x + 16)
Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^5} \, dx=- \frac {15577920 x^{5} + 24665040 x^{4} + 3606000 x^{3} - 10343210 x^{2} - 4966396 x - 460595}{457419312 x^{6} + 762365520 x^{5} + 114354828 x^{4} - 372712032 x^{3} - 146825952 x^{2} + 45177216 x + 22588608} - \frac {24040 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {24040 \log {\left (x + \frac {2}{3} \right )}}{823543} \]
-(15577920*x**5 + 24665040*x**4 + 3606000*x**3 - 10343210*x**2 - 4966396*x - 460595)/(457419312*x**6 + 762365520*x**5 + 114354828*x**4 - 372712032*x **3 - 146825952*x**2 + 45177216*x + 22588608) - 24040*log(x - 1/2)/823543 + 24040*log(x + 2/3)/823543
Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^5} \, dx=-\frac {15577920 \, x^{5} + 24665040 \, x^{4} + 3606000 \, x^{3} - 10343210 \, x^{2} - 4966396 \, x - 460595}{1411788 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} + \frac {24040}{823543} \, \log \left (3 \, x + 2\right ) - \frac {24040}{823543} \, \log \left (2 \, x - 1\right ) \]
-1/1411788*(15577920*x^5 + 24665040*x^4 + 3606000*x^3 - 10343210*x^2 - 496 6396*x - 460595)/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16) + 24040/823543*log(3*x + 2) - 24040/823543*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^5} \, dx=-\frac {5750}{117649 \, {\left (3 \, x + 2\right )}} + \frac {264 \, {\left (\frac {896}{3 \, x + 2} - 223\right )}}{823543 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}^{2}} - \frac {829}{33614 \, {\left (3 \, x + 2\right )}^{2}} + \frac {64}{7203 \, {\left (3 \, x + 2\right )}^{3}} - \frac {1}{1372 \, {\left (3 \, x + 2\right )}^{4}} - \frac {24040}{823543} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]
-5750/117649/(3*x + 2) + 264/823543*(896/(3*x + 2) - 223)/(7/(3*x + 2) - 2 )^2 - 829/33614/(3*x + 2)^2 + 64/7203/(3*x + 2)^3 - 1/1372/(3*x + 2)^4 - 2 4040/823543*log(abs(-7/(3*x + 2) + 2))
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.75 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^5} \, dx=\frac {48080\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}+\frac {-\frac {12020\,x^5}{352947}-\frac {57095\,x^4}{1058841}-\frac {75125\,x^3}{9529569}+\frac {5171605\,x^2}{228709656}+\frac {1241599\,x}{114354828}+\frac {460595}{457419312}}{x^6+\frac {5\,x^5}{3}+\frac {x^4}{4}-\frac {22\,x^3}{27}-\frac {26\,x^2}{81}+\frac {8\,x}{81}+\frac {4}{81}} \]