Integrand size = 22, antiderivative size = 97 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^6 (3+5 x)} \, dx=\frac {64}{1294139 (1-2 x)}+\frac {9}{245 (2+3 x)^5}+\frac {351}{1372 (2+3 x)^4}+\frac {4131}{2401 (2+3 x)^3}+\frac {434043}{33614 (2+3 x)^2}+\frac {15192225}{117649 (2+3 x)}-\frac {14912 \log (1-2 x)}{99648703}-\frac {531729603 \log (2+3 x)}{823543}+\frac {78125}{121} \log (3+5 x) \]
64/1294139/(1-2*x)+9/245/(2+3*x)^5+351/1372/(2+3*x)^4+4131/2401/(2+3*x)^3+ 434043/33614/(2+3*x)^2+15192225/117649/(2+3*x)-14912/99648703*ln(1-2*x)-53 1729603/823543*ln(2+3*x)+78125/121*ln(3+5*x)
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^6 (3+5 x)} \, dx=\frac {\frac {19712}{1-2 x}+\frac {73211292}{5 (2+3 x)^5}+\frac {101972871}{(2+3 x)^4}+\frac {685795572}{(2+3 x)^3}+\frac {5146881894}{(2+3 x)^2}+\frac {51471258300}{2+3 x}-59648 \log (5-10 x)-257357127852 \log (5 (2+3 x))+257357187500 \log (3+5 x)}{398594812} \]
(19712/(1 - 2*x) + 73211292/(5*(2 + 3*x)^5) + 101972871/(2 + 3*x)^4 + 6857 95572/(2 + 3*x)^3 + 5146881894/(2 + 3*x)^2 + 51471258300/(2 + 3*x) - 59648 *Log[5 - 10*x] - 257357127852*Log[5*(2 + 3*x)] + 257357187500*Log[3 + 5*x] )/398594812
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.
\(\displaystyle \int \frac {1}{(1-2 x)^2 (3 x+2)^6 (5 x+3)} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {1595188809}{823543 (3 x+2)}+\frac {390625}{121 (5 x+3)}-\frac {45576675}{117649 (3 x+2)^2}-\frac {1302129}{16807 (3 x+2)^3}-\frac {37179}{2401 (3 x+2)^4}-\frac {1053}{343 (3 x+2)^5}-\frac {27}{49 (3 x+2)^6}-\frac {29824}{99648703 (2 x-1)}+\frac {128}{1294139 (2 x-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {64}{1294139 (1-2 x)}+\frac {15192225}{117649 (3 x+2)}+\frac {434043}{33614 (3 x+2)^2}+\frac {4131}{2401 (3 x+2)^3}+\frac {351}{1372 (3 x+2)^4}+\frac {9}{245 (3 x+2)^5}-\frac {14912 \log (1-2 x)}{99648703}-\frac {531729603 \log (3 x+2)}{823543}+\frac {78125}{121} \log (5 x+3)\) |
64/(1294139*(1 - 2*x)) + 9/(245*(2 + 3*x)^5) + 351/(1372*(2 + 3*x)^4) + 41 31/(2401*(2 + 3*x)^3) + 434043/(33614*(2 + 3*x)^2) + 15192225/(117649*(2 + 3*x)) - (14912*Log[1 - 2*x])/99648703 - (531729603*Log[2 + 3*x])/823543 + (78125*Log[3 + 5*x])/121
3.17.2.3.1 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 = 2.69 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68
method | result | size |
norman | \(\frac {-\frac {220760702913}{25882780} x -\frac {7349232123}{2588278} x^{2}+\frac {27072529398}{1294139} x^{5}+\frac {59559504282}{1294139} x^{4}+\frac {74980599075}{2588278} x^{3}-\frac {28171350293}{12941390}}{\left (-1+2 x \right ) \left (2+3 x \right )^{5}}-\frac {14912 \ln \left (-1+2 x \right )}{99648703}-\frac {531729603 \ln \left (2+3 x \right )}{823543}+\frac {78125 \ln \left (3+5 x \right )}{121}\) | \(66\) |
risch | \(\frac {-\frac {220760702913}{25882780} x -\frac {7349232123}{2588278} x^{2}+\frac {27072529398}{1294139} x^{5}+\frac {59559504282}{1294139} x^{4}+\frac {74980599075}{2588278} x^{3}-\frac {28171350293}{12941390}}{\left (-1+2 x \right ) \left (2+3 x \right )^{5}}-\frac {14912 \ln \left (-1+2 x \right )}{99648703}-\frac {531729603 \ln \left (2+3 x \right )}{823543}+\frac {78125 \ln \left (3+5 x \right )}{121}\) | \(67\) |
default | \(\frac {78125 \ln \left (3+5 x \right )}{121}-\frac {64}{1294139 \left (-1+2 x \right )}-\frac {14912 \ln \left (-1+2 x \right )}{99648703}+\frac {9}{245 \left (2+3 x \right )^{5}}+\frac {351}{1372 \left (2+3 x \right )^{4}}+\frac {4131}{2401 \left (2+3 x \right )^{3}}+\frac {434043}{33614 \left (2+3 x \right )^{2}}+\frac {15192225}{117649 \left (2+3 x \right )}-\frac {531729603 \ln \left (2+3 x \right )}{823543}\) | \(80\) |
parallelrisch | \(-\frac {-109800953181920 x +4941258000000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+7411885282137600 \ln \left (\frac {2}{3}+x \right ) x^{3}+3623589200000000 \ln \left (x +\frac {3}{5}\right ) x -4941256854758400 \ln \left (\frac {2}{3}+x \right ) x^{2}-3623588360156160 \ln \left (\frac {2}{3}+x \right ) x +2319912975849777 x^{5}+1054228270664646 x^{6}-142851150482040 x^{3}+1460865677448870 x^{4}-430064013659280 x^{2}+6441984000 \ln \left (x -\frac {1}{2}\right ) x^{4}+27794569808016000 \ln \left (\frac {2}{3}+x \right ) x^{4}-658834247301120 \ln \left (\frac {2}{3}+x \right )+1717862400 \ln \left (x -\frac {1}{2}\right ) x^{3}-1145241600 \ln \left (x -\frac {1}{2}\right ) x^{2}-839843840 \ln \left (x -\frac {1}{2}\right ) x +658834400000000 \ln \left (x +\frac {3}{5}\right )+28350461204176320 \ln \left (\frac {2}{3}+x \right ) x^{5}-7411887000000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-28350467775000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-27794576250000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+10006045130885760 \ln \left (\frac {2}{3}+x \right ) x^{6}-10006047450000000 \ln \left (x +\frac {3}{5}\right ) x^{6}-152698880 \ln \left (x -\frac {1}{2}\right )+2319114240 \ln \left (x -\frac {1}{2}\right ) x^{6}+6570823680 \ln \left (x -\frac {1}{2}\right ) x^{5}}{31887584960 \left (-1+2 x \right ) \left (2+3 x \right )^{5}}\) | \(220\) |
(-220760702913/25882780*x-7349232123/2588278*x^2+27072529398/1294139*x^5+5 9559504282/1294139*x^4+74980599075/2588278*x^3-28171350293/12941390)/(-1+2 *x)/(2+3*x)^5-14912/99648703*ln(-1+2*x)-531729603/823543*ln(2+3*x)+78125/1 21*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 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^6 (3+5 x)} \, dx=\frac {41691695272920 \, x^{5} + 91721636594280 \, x^{4} + 57735061287750 \, x^{3} - 5658908734710 \, x^{2} + 1286785937500 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (5 \, x + 3\right ) - 1286785639260 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (3 \, x + 2\right ) - 298240 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (2 \, x - 1\right ) - 16998574124301 \, x - 4338387945122}{1992974060 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \]
1/1992974060*(41691695272920*x^5 + 91721636594280*x^4 + 57735061287750*x^3 - 5658908734710*x^2 + 1286785937500*(486*x^6 + 1377*x^5 + 1350*x^4 + 360* x^3 - 240*x^2 - 176*x - 32)*log(5*x + 3) - 1286785639260*(486*x^6 + 1377*x ^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*log(3*x + 2) - 298240*(486 *x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*log(2*x - 1) - 16998574124301*x - 4338387945122)/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x ^3 - 240*x^2 - 176*x - 32)
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^6 (3+5 x)} \, dx=\frac {541450587960 x^{5} + 1191190085640 x^{4} + 749805990750 x^{3} - 73492321230 x^{2} - 220760702913 x - 56342700586}{12579031080 x^{6} + 35640588060 x^{5} + 34941753000 x^{4} + 9317800800 x^{3} - 6211867200 x^{2} - 4555369280 x - 828248960} - \frac {14912 \log {\left (x - \frac {1}{2} \right )}}{99648703} + \frac {78125 \log {\left (x + \frac {3}{5} \right )}}{121} - \frac {531729603 \log {\left (x + \frac {2}{3} \right )}}{823543} \]
(541450587960*x**5 + 1191190085640*x**4 + 749805990750*x**3 - 73492321230* x**2 - 220760702913*x - 56342700586)/(12579031080*x**6 + 35640588060*x**5 + 34941753000*x**4 + 9317800800*x**3 - 6211867200*x**2 - 4555369280*x - 82 8248960) - 14912*log(x - 1/2)/99648703 + 78125*log(x + 3/5)/121 - 53172960 3*log(x + 2/3)/823543
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^6 (3+5 x)} \, dx=\frac {541450587960 \, x^{5} + 1191190085640 \, x^{4} + 749805990750 \, x^{3} - 73492321230 \, x^{2} - 220760702913 \, x - 56342700586}{25882780 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} + \frac {78125}{121} \, \log \left (5 \, x + 3\right ) - \frac {531729603}{823543} \, \log \left (3 \, x + 2\right ) - \frac {14912}{99648703} \, \log \left (2 \, x - 1\right ) \]
1/25882780*(541450587960*x^5 + 1191190085640*x^4 + 749805990750*x^3 - 7349 2321230*x^2 - 220760702913*x - 56342700586)/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32) + 78125/121*log(5*x + 3) - 531729603/82 3543*log(3*x + 2) - 14912/99648703*log(2*x - 1)
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^6 (3+5 x)} \, dx=-\frac {64}{1294139 \, {\left (2 \, x - 1\right )}} - \frac {54 \, {\left (\frac {6617665845}{2 \, x - 1} + \frac {23331909825}{{\left (2 \, x - 1\right )}^{2}} + \frac {36565643625}{{\left (2 \, x - 1\right )}^{3}} + \frac {21492731575}{{\left (2 \, x - 1\right )}^{4}} + 703958526\right )}}{4117715 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{5}} - \frac {531729603}{823543} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) + \frac {78125}{121} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \]
-64/1294139/(2*x - 1) - 54/4117715*(6617665845/(2*x - 1) + 23331909825/(2* x - 1)^2 + 36565643625/(2*x - 1)^3 + 21492731575/(2*x - 1)^4 + 703958526)/ (7/(2*x - 1) + 3)^5 - 531729603/823543*log(abs(-7/(2*x - 1) - 3)) + 78125/ 121*log(abs(-11/(2*x - 1) - 5))
Time = 1.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^6 (3+5 x)} \, dx=\frac {78125\,\ln \left (x+\frac {3}{5}\right )}{121}-\frac {531729603\,\ln \left (x+\frac {2}{3}\right )}{823543}-\frac {14912\,\ln \left (x-\frac {1}{2}\right )}{99648703}-\frac {-\frac {55704793\,x^5}{1294139}-\frac {367651261\,x^4}{3882417}-\frac {2777059225\,x^3}{46589004}+\frac {816581347\,x^2}{139767012}+\frac {73586900971\,x}{4193010360}+\frac {28171350293}{6289515540}}{x^6+\frac {17\,x^5}{6}+\frac {25\,x^4}{9}+\frac {20\,x^3}{27}-\frac {40\,x^2}{81}-\frac {88\,x}{243}-\frac {16}{243}} \]
(78125*log(x + 3/5))/121 - (531729603*log(x + 2/3))/823543 - (14912*log(x - 1/2))/99648703 - ((73586900971*x)/4193010360 + (816581347*x^2)/139767012 - (2777059225*x^3)/46589004 - (367651261*x^4)/3882417 - (55704793*x^5)/12 94139 + 28171350293/6289515540)/((20*x^3)/27 - (40*x^2)/81 - (88*x)/243 + (25*x^4)/9 + (17*x^5)/6 + x^6 - 16/243)