Integrand size = 22, antiderivative size = 97 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {64}{3195731 (1-2 x)}+\frac {27}{49 (2+3 x)^3}+\frac {8829}{686 (2+3 x)^2}+\frac {630342}{2401 (2+3 x)}-\frac {3125}{242 (3+5 x)^2}+\frac {400000}{1331 (3+5 x)}-\frac {15168 \log (1-2 x)}{246071287}-\frac {37214802 \log (2+3 x)}{16807}+\frac {32418750 \log (3+5 x)}{14641} \]
64/3195731/(1-2*x)+27/49/(2+3*x)^3+8829/686/(2+3*x)^2+630342/2401/(2+3*x)- 3125/242/(3+5*x)^2+400000/1331/(3+5*x)-15168/246071287*ln(1-2*x)-37214802/ 16807*ln(2+3*x)+32418750/14641*ln(3+5*x)
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {2 \left (\frac {77}{4} \left (\frac {128}{1-2 x}+\frac {3521826}{(2+3 x)^3}+\frac {82259793}{(2+3 x)^2}+\frac {1677970404}{2+3 x}-\frac {82534375}{(3+5 x)^2}+\frac {1920800000}{3+5 x}\right )-7584 \log (1-2 x)-272430958041 \log (4+6 x)+272430965625 \log (6+10 x)\right )}{246071287} \]
(2*((77*(128/(1 - 2*x) + 3521826/(2 + 3*x)^3 + 82259793/(2 + 3*x)^2 + 1677 970404/(2 + 3*x) - 82534375/(3 + 5*x)^2 + 1920800000/(3 + 5*x)))/4 - 7584* Log[1 - 2*x] - 272430958041*Log[4 + 6*x] + 272430965625*Log[6 + 10*x]))/24 6071287
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.
\(\displaystyle \int \frac {1}{(1-2 x)^2 (3 x+2)^4 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {111644406}{16807 (3 x+2)}+\frac {162093750}{14641 (5 x+3)}-\frac {1891026}{2401 (3 x+2)^2}-\frac {2000000}{1331 (5 x+3)^2}-\frac {26487}{343 (3 x+2)^3}+\frac {15625}{121 (5 x+3)^3}-\frac {243}{49 (3 x+2)^4}-\frac {30336}{246071287 (2 x-1)}+\frac {128}{3195731 (2 x-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {64}{3195731 (1-2 x)}+\frac {630342}{2401 (3 x+2)}+\frac {400000}{1331 (5 x+3)}+\frac {8829}{686 (3 x+2)^2}-\frac {3125}{242 (5 x+3)^2}+\frac {27}{49 (3 x+2)^3}-\frac {15168 \log (1-2 x)}{246071287}-\frac {37214802 \log (3 x+2)}{16807}+\frac {32418750 \log (5 x+3)}{14641}\) |
64/(3195731*(1 - 2*x)) + 27/(49*(2 + 3*x)^3) + 8829/(686*(2 + 3*x)^2) + 63 0342/(2401*(2 + 3*x)) - 3125/(242*(3 + 5*x)^2) + 400000/(1331*(3 + 5*x)) - (15168*Log[1 - 2*x])/246071287 - (37214802*Log[2 + 3*x])/16807 + (3241875 0*Log[3 + 5*x])/14641
3.17.29.3.1 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 = 0.88 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75
method | result | size |
norman | \(\frac {-\frac {456430279071}{6391462} x -\frac {114775016133}{3195731} x^{2}+\frac {636851297700}{3195731} x^{5}+\frac {754550478837}{3195731} x^{3}+\frac {1316159177940}{3195731} x^{4}-\frac {53679120734}{3195731}}{\left (-1+2 x \right ) \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-\frac {15168 \ln \left (-1+2 x \right )}{246071287}-\frac {37214802 \ln \left (2+3 x \right )}{16807}+\frac {32418750 \ln \left (3+5 x \right )}{14641}\) | \(73\) |
risch | \(\frac {-\frac {456430279071}{6391462} x -\frac {114775016133}{3195731} x^{2}+\frac {636851297700}{3195731} x^{5}+\frac {754550478837}{3195731} x^{3}+\frac {1316159177940}{3195731} x^{4}-\frac {53679120734}{3195731}}{\left (-1+2 x \right ) \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-\frac {15168 \ln \left (-1+2 x \right )}{246071287}-\frac {37214802 \ln \left (2+3 x \right )}{16807}+\frac {32418750 \ln \left (3+5 x \right )}{14641}\) | \(74\) |
default | \(-\frac {3125}{242 \left (3+5 x \right )^{2}}+\frac {400000}{1331 \left (3+5 x \right )}+\frac {32418750 \ln \left (3+5 x \right )}{14641}-\frac {64}{3195731 \left (-1+2 x \right )}-\frac {15168 \ln \left (-1+2 x \right )}{246071287}+\frac {27}{49 \left (2+3 x \right )^{3}}+\frac {8829}{686 \left (2+3 x \right )^{2}}+\frac {630342}{2401 \left (2+3 x \right )}-\frac {37214802 \ln \left (2+3 x \right )}{16807}\) | \(80\) |
parallelrisch | \(-\frac {-470758030952748 x +25028777673900000 \ln \left (x +\frac {3}{5}\right ) x^{2}+30167914569628176 \ln \left (\frac {2}{3}+x \right ) x^{3}+16476624801000000 \ln \left (x +\frac {3}{5}\right ) x -25028776977142752 \ln \left (\frac {2}{3}+x \right ) x^{2}-16476624342319680 \ln \left (\frac {2}{3}+x \right ) x +11535146826359310 x^{5}+5579944600299300 x^{6}-1004726078649986 x^{3}+6615875387580228 x^{4}-2000727795737132 x^{2}+3675995136 \ln \left (x -\frac {1}{2}\right ) x^{4}+132048375086304864 \ln \left (\frac {2}{3}+x \right ) x^{4}-2824564172969088 \ln \left (\frac {2}{3}+x \right )+839821824 \ln \left (x -\frac {1}{2}\right ) x^{3}-696757248 \ln \left (x -\frac {1}{2}\right ) x^{2}-458680320 \ln \left (x -\frac {1}{2}\right ) x +2824564251600000 \ln \left (x +\frac {3}{5}\right )+142993561256560080 \ln \left (\frac {2}{3}+x \right ) x^{5}-30167915409450000 \ln \left (x +\frac {3}{5}\right ) x^{3}-142993565237250000 \ln \left (x +\frac {3}{5}\right ) x^{5}-132048378762300000 \ln \left (x +\frac {3}{5}\right ) x^{4}+52960578243170400 \ln \left (\frac {2}{3}+x \right ) x^{6}-52960579717500000 \ln \left (x +\frac {3}{5}\right ) x^{6}-78630912 \ln \left (x -\frac {1}{2}\right )+1474329600 \ln \left (x -\frac {1}{2}\right ) x^{6}+3980689920 \ln \left (x -\frac {1}{2}\right ) x^{5}}{17717132664 \left (-1+2 x \right ) \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}\) | \(227\) |
(-456430279071/6391462*x-114775016133/3195731*x^2+636851297700/3195731*x^5 +754550478837/3195731*x^3+1316159177940/3195731*x^4-53679120734/3195731)/( -1+2*x)/(2+3*x)^3/(3+5*x)^2-15168/246071287*ln(-1+2*x)-37214802/16807*ln(2 +3*x)+32418750/14641*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 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {98075099845800 \, x^{5} + 202688513402760 \, x^{4} + 116200773740898 \, x^{3} - 17675352484482 \, x^{2} + 1089723862500 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )} \log \left (5 \, x + 3\right ) - 1089723832164 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )} \log \left (3 \, x + 2\right ) - 30336 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )} \log \left (2 \, x - 1\right ) - 35145131488467 \, x - 8266584593036}{492142574 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )}} \]
1/492142574*(98075099845800*x^5 + 202688513402760*x^4 + 116200773740898*x^ 3 - 17675352484482*x^2 + 1089723862500*(1350*x^6 + 3645*x^5 + 3366*x^4 + 7 69*x^3 - 638*x^2 - 420*x - 72)*log(5*x + 3) - 1089723832164*(1350*x^6 + 36 45*x^5 + 3366*x^4 + 769*x^3 - 638*x^2 - 420*x - 72)*log(3*x + 2) - 30336*( 1350*x^6 + 3645*x^5 + 3366*x^4 + 769*x^3 - 638*x^2 - 420*x - 72)*log(2*x - 1) - 35145131488467*x - 8266584593036)/(1350*x^6 + 3645*x^5 + 3366*x^4 + 769*x^3 - 638*x^2 - 420*x - 72)
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {1273702595400 x^{5} + 2632318355880 x^{4} + 1509100957674 x^{3} - 229550032266 x^{2} - 456430279071 x - 107358241468}{8628473700 x^{6} + 23296878990 x^{5} + 21513661092 x^{4} + 4915034278 x^{3} - 4077752756 x^{2} - 2684414040 x - 460185264} - \frac {15168 \log {\left (x - \frac {1}{2} \right )}}{246071287} + \frac {32418750 \log {\left (x + \frac {3}{5} \right )}}{14641} - \frac {37214802 \log {\left (x + \frac {2}{3} \right )}}{16807} \]
(1273702595400*x**5 + 2632318355880*x**4 + 1509100957674*x**3 - 2295500322 66*x**2 - 456430279071*x - 107358241468)/(8628473700*x**6 + 23296878990*x* *5 + 21513661092*x**4 + 4915034278*x**3 - 4077752756*x**2 - 2684414040*x - 460185264) - 15168*log(x - 1/2)/246071287 + 32418750*log(x + 3/5)/14641 - 37214802*log(x + 2/3)/16807
Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {1273702595400 \, x^{5} + 2632318355880 \, x^{4} + 1509100957674 \, x^{3} - 229550032266 \, x^{2} - 456430279071 \, x - 107358241468}{6391462 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )}} + \frac {32418750}{14641} \, \log \left (5 \, x + 3\right ) - \frac {37214802}{16807} \, \log \left (3 \, x + 2\right ) - \frac {15168}{246071287} \, \log \left (2 \, x - 1\right ) \]
1/6391462*(1273702595400*x^5 + 2632318355880*x^4 + 1509100957674*x^3 - 229 550032266*x^2 - 456430279071*x - 107358241468)/(1350*x^6 + 3645*x^5 + 3366 *x^4 + 769*x^3 - 638*x^2 - 420*x - 72) + 32418750/14641*log(5*x + 3) - 372 14802/16807*log(3*x + 2) - 15168/246071287*log(2*x - 1)
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^3} \, dx=-\frac {64}{3195731 \, {\left (2 \, x - 1\right )}} - \frac {4 \, {\left (\frac {49415890344165}{2 \, x - 1} + \frac {169212487575969}{{\left (2 \, x - 1\right )}^{2}} + \frac {257446971133345}{{\left (2 \, x - 1\right )}^{3}} + \frac {146840081089779}{{\left (2 \, x - 1\right )}^{4}} + 5410112162850\right )}}{246071287 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2} {\left (\frac {7}{2 \, x - 1} + 3\right )}^{3}} - \frac {37214802}{16807} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) + \frac {32418750}{14641} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \]
-64/3195731/(2*x - 1) - 4/246071287*(49415890344165/(2*x - 1) + 1692124875 75969/(2*x - 1)^2 + 257446971133345/(2*x - 1)^3 + 146840081089779/(2*x - 1 )^4 + 5410112162850)/((11/(2*x - 1) + 5)^2*(7/(2*x - 1) + 3)^3) - 37214802 /16807*log(abs(-7/(2*x - 1) - 3)) + 32418750/14641*log(abs(-11/(2*x - 1) - 5))
Time = 1.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {32418750\,\ln \left (x+\frac {3}{5}\right )}{14641}-\frac {37214802\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {15168\,\ln \left (x-\frac {1}{2}\right )}{246071287}-\frac {-\frac {471741702\,x^5}{3195731}-\frac {4874663622\,x^4}{15978655}-\frac {27946314031\,x^3}{159786550}+\frac {38258338711\,x^2}{1438078950}+\frac {152143426357\,x}{2876157900}+\frac {26839560367}{2157118425}}{x^6+\frac {27\,x^5}{10}+\frac {187\,x^4}{75}+\frac {769\,x^3}{1350}-\frac {319\,x^2}{675}-\frac {14\,x}{45}-\frac {4}{75}} \]
(32418750*log(x + 3/5))/14641 - (37214802*log(x + 2/3))/16807 - (15168*log (x - 1/2))/246071287 - ((152143426357*x)/2876157900 + (38258338711*x^2)/14 38078950 - (27946314031*x^3)/159786550 - (4874663622*x^4)/15978655 - (4717 41702*x^5)/3195731 + 26839560367/2157118425)/((769*x^3)/1350 - (319*x^2)/6 75 - (14*x)/45 + (187*x^4)/75 + (27*x^5)/10 + x^6 - 4/75)