Integrand size = 22, antiderivative size = 97 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^3} \, dx=\frac {27}{28 (2+3 x)^4}+\frac {963}{49 (2+3 x)^3}+\frac {102114}{343 (2+3 x)^2}+\frac {11984706}{2401 (2+3 x)}-\frac {3125}{22 (3+5 x)^2}+\frac {509375}{121 (3+5 x)}-\frac {128 \log (1-2 x)}{22370117}-\frac {631722537 \log (2+3 x)}{16807}+\frac {50028125 \log (3+5 x)}{1331} \]
27/28/(2+3*x)^4+963/49/(2+3*x)^3+102114/343/(2+3*x)^2+11984706/2401/(2+3*x )-3125/22/(3+5*x)^2+509375/121/(3+5*x)-128/22370117*ln(1-2*x)-631722537/16 807*ln(2+3*x)+50028125/1331*ln(3+5*x)
Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^3} \, dx=\frac {27}{28 (2+3 x)^4}+\frac {963}{49 (2+3 x)^3}+\frac {102114}{343 (2+3 x)^2}+\frac {11984706}{2401 (2+3 x)}-\frac {3125}{22 (3+5 x)^2}+\frac {509375}{363+605 x}-\frac {128 \log (1-2 x)}{22370117}-\frac {631722537 \log (4+6 x)}{16807}+\frac {50028125 \log (6+10 x)}{1331} \]
27/(28*(2 + 3*x)^4) + 963/(49*(2 + 3*x)^3) + 102114/(343*(2 + 3*x)^2) + 11 984706/(2401*(2 + 3*x)) - 3125/(22*(3 + 5*x)^2) + 509375/(363 + 605*x) - ( 128*Log[1 - 2*x])/22370117 - (631722537*Log[4 + 6*x])/16807 + (50028125*Lo g[6 + 10*x])/1331
Time = 0.23 (sec) , antiderivative size = 97, 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.
| \(\displaystyle \int \frac {1}{(1-2 x) (3 x+2)^5 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {1895167611}{16807 (3 x+2)}+\frac {250140625}{1331 (5 x+3)}-\frac {35954118}{2401 (3 x+2)^2}-\frac {2546875}{121 (5 x+3)^2}-\frac {612684}{343 (3 x+2)^3}+\frac {15625}{11 (5 x+3)^3}-\frac {8667}{49 (3 x+2)^4}-\frac {81}{7 (3 x+2)^5}-\frac {256}{22370117 (2 x-1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {11984706}{2401 (3 x+2)}+\frac {509375}{121 (5 x+3)}+\frac {102114}{343 (3 x+2)^2}-\frac {3125}{22 (5 x+3)^2}+\frac {963}{49 (3 x+2)^3}+\frac {27}{28 (3 x+2)^4}-\frac {128 \log (1-2 x)}{22370117}-\frac {631722537 \log (3 x+2)}{16807}+\frac {50028125 \log (5 x+3)}{1331}\) |
27/(28*(2 + 3*x)^4) + 963/(49*(2 + 3*x)^3) + 102114/(343*(2 + 3*x)^2) + 11 984706/(2401*(2 + 3*x)) - 3125/(22*(3 + 5*x)^2) + 509375/(121*(3 + 5*x)) - (128*Log[1 - 2*x])/22370117 - (631722537*Log[2 + 3*x])/16807 + (50028125* Log[3 + 5*x])/1331
3.16.29.3.1 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 = 2.61 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68
| method | result | size |
| norman | \(\frac {\frac {1474169659425}{290521} x^{5}+\frac {2568430130789}{581042} x +\frac {6160937191623}{290521} x^{3}+\frac {9532963793295}{581042} x^{4}+\frac {15916809968421}{1162084} x^{2}+\frac {662695553413}{1162084}}{\left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-\frac {128 \ln \left (-1+2 x \right )}{22370117}-\frac {631722537 \ln \left (2+3 x \right )}{16807}+\frac {50028125 \ln \left (3+5 x \right )}{1331}\) | \(66\) |
| risch | \(\frac {\frac {1474169659425}{290521} x^{5}+\frac {2568430130789}{581042} x +\frac {6160937191623}{290521} x^{3}+\frac {9532963793295}{581042} x^{4}+\frac {15916809968421}{1162084} x^{2}+\frac {662695553413}{1162084}}{\left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-\frac {128 \ln \left (-1+2 x \right )}{22370117}-\frac {631722537 \ln \left (2+3 x \right )}{16807}+\frac {50028125 \ln \left (3+5 x \right )}{1331}\) | \(67\) |
| default | \(-\frac {3125}{22 \left (3+5 x \right )^{2}}+\frac {509375}{121 \left (3+5 x \right )}+\frac {50028125 \ln \left (3+5 x \right )}{1331}-\frac {128 \ln \left (-1+2 x \right )}{22370117}+\frac {27}{28 \left (2+3 x \right )^{4}}+\frac {963}{49 \left (2+3 x \right )^{3}}+\frac {102114}{343 \left (2+3 x \right )^{2}}+\frac {11984706}{2401 \left (2+3 x \right )}-\frac {631722537 \ln \left (2+3 x \right )}{16807}\) | \(80\) |
| parallelrisch | \(-\frac {11623530851227680 x -2530055674641600000 \ln \left (x +\frac {3}{5}\right ) x^{2}+5242213364883568128 \ln \left (\frac {2}{3}+x \right ) x^{3}-650917845849600000 \ln \left (x +\frac {3}{5}\right ) x +2530055674256444928 \ln \left (\frac {2}{3}+x \right ) x^{2}+650917845750509568 \ln \left (\frac {2}{3}+x \right ) x +334163403373414230 x^{5}+103330804165922025 x^{6}+279072397278094728 x^{3}+432003468859697889 x^{4}+90082372039420376 x^{2}+929636352 \ln \left (x -\frac {1}{2}\right ) x^{4}+6106713628770963648 \ln \left (\frac {2}{3}+x \right ) x^{4}+69741197758983168 \ln \left (\frac {2}{3}+x \right )+798031872 \ln \left (x -\frac {1}{2}\right ) x^{3}+385155072 \ln \left (x -\frac {1}{2}\right ) x^{2}+99090432 \ln \left (x -\frac {1}{2}\right ) x -69741197769600000 \ln \left (x +\frac {3}{5}\right )+3792177628144709760 \ln \left (\frac {2}{3}+x \right ) x^{5}-5242213365681600000 \ln \left (x +\frac {3}{5}\right ) x^{3}-3792177628722000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-6106713629700600000 \ln \left (x +\frac {3}{5}\right ) x^{4}+980735593485700800 \ln \left (\frac {2}{3}+x \right ) x^{6}-980735593635000000 \ln \left (x +\frac {3}{5}\right ) x^{6}+10616832 \ln \left (x -\frac {1}{2}\right )+149299200 \ln \left (x -\frac {1}{2}\right ) x^{6}+577290240 \ln \left (x -\frac {1}{2}\right ) x^{5}}{12885187392 \left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}\) | \(220\) |
(1474169659425/290521*x^5+2568430130789/581042*x+6160937191623/290521*x^3+ 9532963793295/581042*x^4+15916809968421/1162084*x^2+662695553413/1162084)/ (2+3*x)^4/(3+5*x)^2-128/22370117*ln(-1+2*x)-631722537/16807*ln(2+3*x)+5002 8125/1331*ln(3+5*x)
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (79) = 158\).
Time = 0.22 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.78 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^3} \, dx=\frac {454044255102900 \, x^{5} + 1468076424167430 \, x^{4} + 1897568655019884 \, x^{3} + 1225594367568417 \, x^{2} + 3363290787500 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (5 \, x + 3\right ) - 3363290786988 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (3 \, x + 2\right ) - 512 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (2 \, x - 1\right ) + 395538240141506 \, x + 51027557612801}{89480468 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]
1/89480468*(454044255102900*x^5 + 1468076424167430*x^4 + 1897568655019884* x^3 + 1225594367568417*x^2 + 3363290787500*(2025*x^6 + 7830*x^5 + 12609*x^ 4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log(5*x + 3) - 3363290786988*(202 5*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log(3* x + 2) - 512*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 134 4*x + 144)*log(2*x - 1) + 395538240141506*x + 51027557612801)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)
Time = 0.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^3} \, dx=- \frac {- 5896678637700 x^{5} - 19065927586590 x^{4} - 24643748766492 x^{3} - 15916809968421 x^{2} - 5136860261578 x - 662695553413}{2353220100 x^{6} + 9099117720 x^{5} + 14652717156 x^{4} + 12578397216 x^{3} + 6070726816 x^{2} + 1561840896 x + 167340096} - \frac {128 \log {\left (x - \frac {1}{2} \right )}}{22370117} + \frac {50028125 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {631722537 \log {\left (x + \frac {2}{3} \right )}}{16807} \]
-(-5896678637700*x**5 - 19065927586590*x**4 - 24643748766492*x**3 - 159168 09968421*x**2 - 5136860261578*x - 662695553413)/(2353220100*x**6 + 9099117 720*x**5 + 14652717156*x**4 + 12578397216*x**3 + 6070726816*x**2 + 1561840 896*x + 167340096) - 128*log(x - 1/2)/22370117 + 50028125*log(x + 3/5)/133 1 - 631722537*log(x + 2/3)/16807
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^3} \, dx=\frac {5896678637700 \, x^{5} + 19065927586590 \, x^{4} + 24643748766492 \, x^{3} + 15916809968421 \, x^{2} + 5136860261578 \, x + 662695553413}{1162084 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} + \frac {50028125}{1331} \, \log \left (5 \, x + 3\right ) - \frac {631722537}{16807} \, \log \left (3 \, x + 2\right ) - \frac {128}{22370117} \, \log \left (2 \, x - 1\right ) \]
1/1162084*(5896678637700*x^5 + 19065927586590*x^4 + 24643748766492*x^3 + 1 5916809968421*x^2 + 5136860261578*x + 662695553413)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144) + 50028125/1331*log(5*x + 3) - 631722537/16807*log(3*x + 2) - 128/22370117*log(2*x - 1)
Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^3} \, dx=\frac {11984706}{2401 \, {\left (3 \, x + 2\right )}} - \frac {46875 \, {\left (\frac {392}{3 \, x + 2} - 1795\right )}}{242 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {102114}{343 \, {\left (3 \, x + 2\right )}^{2}} + \frac {963}{49 \, {\left (3 \, x + 2\right )}^{3}} + \frac {27}{28 \, {\left (3 \, x + 2\right )}^{4}} + \frac {50028125}{1331} \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) - \frac {128}{22370117} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]
11984706/2401/(3*x + 2) - 46875/242*(392/(3*x + 2) - 1795)/(1/(3*x + 2) - 5)^2 + 102114/343/(3*x + 2)^2 + 963/49/(3*x + 2)^3 + 27/28/(3*x + 2)^4 + 5 0028125/1331*log(abs(-1/(3*x + 2) + 5)) - 128/22370117*log(abs(-7/(3*x + 2 ) + 2))
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^3} \, dx=\frac {50028125\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {631722537\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {128\,\ln \left (x-\frac {1}{2}\right )}{22370117}+\frac {\frac {727985017\,x^5}{290521}+\frac {70614546617\,x^4}{8715630}+\frac {76060952983\,x^3}{7263025}+\frac {5305603322807\,x^2}{784406700}+\frac {2568430130789\,x}{1176610050}+\frac {662695553413}{2353220100}}{x^6+\frac {58\,x^5}{15}+\frac {467\,x^4}{75}+\frac {3608\,x^3}{675}+\frac {5224\,x^2}{2025}+\frac {448\,x}{675}+\frac {16}{225}} \]
(50028125*log(x + 3/5))/1331 - (631722537*log(x + 2/3))/16807 - (128*log(x - 1/2))/22370117 + ((2568430130789*x)/1176610050 + (5305603322807*x^2)/78 4406700 + (76060952983*x^3)/7263025 + (70614546617*x^4)/8715630 + (7279850 17*x^5)/290521 + 662695553413/2353220100)/((448*x)/675 + (5224*x^2)/2025 + (3608*x^3)/675 + (467*x^4)/75 + (58*x^5)/15 + x^6 + 16/225)