\(\int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^3} \, dx\) [474]

Optimal result

Integrand size = 22, antiderivative size = 108 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {32}{3195731 (1-2 x)^2}+\frac {15168}{246071287 (1-2 x)}+\frac {81}{343 (2+3 x)^3}+\frac {26973}{4802 (2+3 x)^2}+\frac {1944972}{16807 (2+3 x)}-\frac {15625}{2662 (3+5 x)^2}+\frac {1968750}{14641 (3+5 x)}-\frac {2054400 \log (1-2 x)}{18947489099}-\frac {115534350 \log (2+3 x)}{117649}+\frac {158156250 \log (3+5 x)}{161051} \] Output:

32/3195731/(1-2*x)^2+15168/(246071287-492142574*x)+81/343/(2+3*x)^3+26973/ 
4802/(2+3*x)^2+1944972/(33614+50421*x)-15625/2662/(3+5*x)^2+1968750/(43923 
+73205*x)-2054400/18947489099*ln(1-2*x)-115534350/117649*ln(2+3*x)+1581562 
50/161051*ln(3+5*x)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^3} \, dx=-\frac {3 \left (-\frac {77 \left (3666255393392+8254486652965 x-23334840827100 x^2-67213599053550 x^3+13177709631900 x^4+136289326113000 x^5+86993245890000 x^6\right )}{3 (2+3 x)^3 \left (-3+x+10 x^2\right )^2}+1369600 \log (3-6 x)[phone] \log (2+3 x)-12404616437500 \log (-3 (3+5 x))\right )}{37894978198} \] Input:

Integrate[1/((1 - 2*x)^3*(2 + 3*x)^4*(3 + 5*x)^3),x]
 

Output:

(-3*((-77*(3666255393392 + 8254486652965*x - 23334840827100*x^2 - 67213599 
053550*x^3 + 13177709631900*x^4 + 136289326113000*x^5 + 86993245890000*x^6 
))/(3*(2 + 3*x)^3*(-3 + x + 10*x^2)^2) + 1369600*Log[3 - 6*x] + 1240461506 
7900*Log[2 + 3*x] - 12404616437500*Log[-3*(3 + 5*x)]))/37894978198
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 108, 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.

\(\displaystyle \int \frac {1}{(1-2 x)^3 (3 x+2)^4 (5 x+3)^3} \, dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {346603050}{117649 (3 x+2)}+\frac {790781250}{161051 (5 x+3)}-\frac {5834916}{16807 (3 x+2)^2}-\frac {9843750}{14641 (5 x+3)^2}-\frac {80919}{2401 (3 x+2)^3}+\frac {78125}{1331 (5 x+3)^3}-\frac {729}{343 (3 x+2)^4}-\frac {4108800}{18947489099 (2 x-1)}+\frac {30336}{246071287 (2 x-1)^2}-\frac {128}{3195731 (2 x-1)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {15168}{246071287 (1-2 x)}+\frac {1944972}{16807 (3 x+2)}+\frac {1968750}{14641 (5 x+3)}+\frac {32}{3195731 (1-2 x)^2}+\frac {26973}{4802 (3 x+2)^2}-\frac {15625}{2662 (5 x+3)^2}+\frac {81}{343 (3 x+2)^3}-\frac {2054400 \log (1-2 x)}{18947489099}-\frac {115534350 \log (3 x+2)}{117649}+\frac {158156250 \log (5 x+3)}{161051}\)

Input:

Int[1/((1 - 2*x)^3*(2 + 3*x)^4*(3 + 5*x)^3),x]
 

Output:

32/(3195731*(1 - 2*x)^2) + 15168/(246071287*(1 - 2*x)) + 81/(343*(2 + 3*x) 
^3) + 26973/(4802*(2 + 3*x)^2) + 1944972/(16807*(2 + 3*x)) - 15625/(2662*( 
3 + 5*x)^2) + 1968750/(14641*(3 + 5*x)) - (2054400*Log[1 - 2*x])/189474890 
99 - (115534350*Log[2 + 3*x])/117649 + (158156250*Log[3 + 5*x])/161051
 

Defintions of rubi rules used

rule 99

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]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72

method	result	size
norman	\(\frac {-\frac {33606799526775}{246071287} x^{3}-\frac {1060674583050}{22370117} x^{2}+\frac {3437936965}{204974} x +\frac {598986801450}{22370117} x^{4}+\frac {43496622945000}{246071287} x^{6}+\frac {68144663056500}{246071287} x^{5}+\frac {1833127696696}{246071287}}{\left (3+5 x \right )^{2} \left (-1+2 x \right )^{2} \left (2+3 x \right )^{3}}-\frac {2054400 \ln \left (-1+2 x \right )}{18947489099}-\frac {115534350 \ln \left (2+3 x \right )}{117649}+\frac {158156250 \ln \left (3+5 x \right )}{161051}\)	\(78\)
risch	\(\frac {-\frac {33606799526775}{246071287} x^{3}-\frac {1060674583050}{22370117} x^{2}+\frac {3437936965}{204974} x +\frac {598986801450}{22370117} x^{4}+\frac {43496622945000}{246071287} x^{6}+\frac {68144663056500}{246071287} x^{5}+\frac {1833127696696}{246071287}}{\left (3+5 x \right )^{2} \left (-1+2 x \right )^{2} \left (2+3 x \right )^{3}}-\frac {2054400 \ln \left (-1+2 x \right )}{18947489099}-\frac {115534350 \ln \left (2+3 x \right )}{117649}+\frac {158156250 \ln \left (3+5 x \right )}{161051}\)	\(79\)
default	\(-\frac {15625}{2662 \left (3+5 x \right )^{2}}+\frac {1968750}{14641 \left (3+5 x \right )}+\frac {158156250 \ln \left (3+5 x \right )}{161051}+\frac {81}{343 \left (2+3 x \right )^{3}}+\frac {26973}{4802 \left (2+3 x \right )^{2}}+\frac {1944972}{16807 \left (2+3 x \right )}-\frac {115534350 \ln \left (2+3 x \right )}{117649}+\frac {32}{3195731 \left (-1+2 x \right )^{2}}-\frac {15168}{246071287 \left (-1+2 x \right )}-\frac {2054400 \ln \left (-1+2 x \right )}{18947489099}\)	\(89\)
parallelrisch	\(-\frac {16076192808164412 x +381107248143098400 x^{7}+270619112200500000 \ln \left (x +\frac {3}{5}\right ) x^{2}-102337356183795040 x^{3}-294552333175768976 x^{4}+36171710578311616 x^{2}-302489856000 \ln \left (x -\frac {1}{2}\right ) x^{3}[phone] \ln \left (x -\frac {1}{2}\right )+57938608391706504 x^{5}+597290668307736480 x^{6}+3617185753799640000 \ln \left (\frac {2}{3}+x \right ) x^{7}-3617186153175000000 \ln \left (x +\frac {3}{5}\right ) x^{7}+4135649045177588400 \ln \left (\frac {2}{3}+x \right ) x^{5}+2739683586386250000 \ln \left (x +\frac {3}{5}\right ) x^{3}-2739683283896394000 \ln \left (\frac {2}{3}+x \right ) x^{3}+96458286767990400 \ln \left (\frac {2}{3}+x \right )+369756765943963200 \ln \left (\frac {2}{3}+x \right ) x -270619082321306400 \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (x -\frac {1}{2}\right ) x^{5}+7957808658359208000 \ln \left (\frac {2}{3}+x \right ) x^{6}-7957809536985000000 \ln \left (x +\frac {3}{5}\right ) x^{6}-369756806769000000 \ln \left (x +\frac {3}{5}\right ) x -2448968725165089600 \ln \left (\frac {2}{3}+x \right ) x^{4}-270391910400 \ln \left (x -\frac {1}{2}\right ) x^{4}+2448968995557000000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (x -\frac {1}{2}\right ) x^{7}-4135649501796750000 \ln \left (x +\frac {3}{5}\right ) x^{5}-29879193600 \ln \left (x -\frac {1}{2}\right ) x^{2}-96458297418000000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (x -\frac {1}{2}\right ) x^{6}[phone] \ln \left (x -\frac {1}{2}\right ) x}{1364219215128 \left (3+5 x \right )^{2} \left (-1+2 x \right )^{2} \left (2+3 x \right )^{3}}\)	\(259\)

Input:

int(1/(1-2*x)^3/(2+3*x)^4/(3+5*x)^3,x,method=_RETURNVERBOSE)
 

Output:

(-33606799526775/246071287*x^3-1060674583050/22370117*x^2+3437936965/20497 
4*x[phone]/22370117*x^4+43496622945000/246071287*x^6+68144663056500/ 
246071287*x^5+1833127696696/246071287)/(3+5*x)^2/(-1+2*x)^2/(2+3*x)^3-2054 
400/18947489099*ln(-1+2*x)-115534350/117649*ln(2+3*x)[phone]/161051*ln( 
3+5*x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (88) = 176\).

Time = 0.07 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.83 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {6698479933530000 \, x^{6} + 10494278110701000 \, x^{5} + 1014683641656300 \, x^{4} - 5175447127123350 \, x^{3} - 1796782743686700 \, x^{2} + 37213849312500 \, {\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 37213845203700 \, {\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )} \log \left (3 \, x + 2\right ) - 4108800 \, {\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )} \log \left (2 \, x - 1\right ) + 635595472278305 \, x + 282301665291184}{37894978198 \, {\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )}} \] Input:

integrate(1/(1-2*x)^3/(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")
 

Output:

1/37894978198*(6698479933530000*x^6 + 10494278110701000*x^5 + 101468364165 
6300*x^4 - 5175447127123350*x^3 - 1796782743686700*x^2 + 37213849312500*(2 
700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 276*x + 72 
)*log(5*x + 3) - 37213845203700*(2700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 
 - 2045*x^3 - 202*x^2 + 276*x + 72)*log(3*x + 2) - 4108800*(2700*x^7 + 594 
0*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 276*x + 72)*log(2*x - 1 
) + 635595472278305*x + 282301665291184)/(2700*x^7 + 5940*x^6 + 3087*x^5 - 
 1828*x^4 - 2045*x^3 - 202*x^2 + 276*x + 72)
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^3} \, dx=- \frac {- 86993245890000 x^{6} - 136289326113000 x^{5} - 13177709631900 x^{4} + 67213599053550 x^{3} + 23334840827100 x^{2} - 8254486652965 x - 3666255393392}{1328784949800 x^{7} + 2923326889560 x^{6} + 1519244125938 x^{5} - 899636625272 x^{4} - 1006431563830 x^{3} - 99412799948 x^{2} + 135831350424 x + 35434265328} - \frac {2054400 \log {\left (x - \frac {1}{2} \right )}}{18947489099} + \frac {158156250 \log {\left (x + \frac {3}{5} \right )}}{161051} - \frac {115534350 \log {\left (x + \frac {2}{3} \right )}}{117649} \] Input:

integrate(1/(1-2*x)**3/(2+3*x)**4/(3+5*x)**3,x)
 

Output:

-(-86993245890000*x**6 - 136289326113000*x**5 - 13177709631900*x**4 + 6721 
3599053550*x**3 + 23334840827100*x**2 - 8254486652965*x - 3666255393392)/( 
1328784949800*x**7 + 2923326889560*x**6 + 1519244125938*x**5 - 89963662527 
2*x**4 - 1006431563830*x**3 - 99412799948*x**2 + 135831350424*x + 35434265 
328) - 2054400*log(x - 1/2)/18947489099 + 158156250*log(x + 3/5)/161051 - 
115534350*log(x + 2/3)/117649
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {86993245890000 \, x^{6} + 136289326113000 \, x^{5} + 13177709631900 \, x^{4} - 67213599053550 \, x^{3} - 23334840827100 \, x^{2} + 8254486652965 \, x + 3666255393392}{492142574 \, {\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )}} + \frac {158156250}{161051} \, \log \left (5 \, x + 3\right ) - \frac {115534350}{117649} \, \log \left (3 \, x + 2\right ) - \frac {2054400}{18947489099} \, \log \left (2 \, x - 1\right ) \] Input:

integrate(1/(1-2*x)^3/(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")
 

Output:

1/492142574*(86993245890000*x^6 + 136289326113000*x^5 + 13177709631900*x^4 
 - 67213599053550*x^3 - 23334840827100*x^2 + 8254486652965*x + 36662553933 
92)/(2700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 276* 
x + 72) + 158156250/161051*log(5*x + 3) - 115534350/117649*log(3*x + 2) - 
2054400/18947489099*log(2*x - 1)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {86993245890000 \, x^{6} + 136289326113000 \, x^{5} + 13177709631900 \, x^{4} - 67213599053550 \, x^{3} - 23334840827100 \, x^{2} + 8254486652965 \, x + 3666255393392}{492142574 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{3} {\left (2 \, x - 1\right )}^{2}} + \frac {158156250}{161051} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {115534350}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {2054400}{18947489099} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \] Input:

integrate(1/(1-2*x)^3/(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")
 

Output:

1/492142574*(86993245890000*x^6 + 136289326113000*x^5 + 13177709631900*x^4 
 - 67213599053550*x^3 - 23334840827100*x^2 + 8254486652965*x + 36662553933 
92)/((5*x + 3)^2*(3*x + 2)^3*(2*x - 1)^2) + 158156250/161051*log(abs(5*x + 
 3)) - 115534350/117649*log(abs(3*x + 2)) - 2054400/18947489099*log(abs(2* 
x - 1))
 

Mupad [B] (verification not implemented)

Time = 1.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {158156250\,\ln \left (x+\frac {3}{5}\right )}{161051}-\frac {115534350\,\ln \left (x+\frac {2}{3}\right )}{117649}-\frac {2054400\,\ln \left (x-\frac {1}{2}\right )}{18947489099}+\frac {\frac {16109860350\,x^6}{246071287}+\frac {25238764095\,x^5}{246071287}+\frac {443693927\,x^4}{44740234}-\frac {448090660357\,x^3}{8858566332}-\frac {2357054629\,x^2}{134220702}+\frac {687587393\,x}{110685960}+\frac {458281924174}{166098118725}}{x^7+\frac {11\,x^6}{5}+\frac {343\,x^5}{300}-\frac {457\,x^4}{675}-\frac {409\,x^3}{540}-\frac {101\,x^2}{1350}+\frac {23\,x}{225}+\frac {2}{75}} \] Input:

int(-1/((2*x - 1)^3*(3*x + 2)^4*(5*x + 3)^3),x)
 

Output:

(158156250*log(x + 3/5))/161051 - (115534350*log(x + 2/3))/117649 - (20544 
00*log(x - 1/2))/18947489099 + ((687587393*x)/110685960 - (2357054629*x^2) 
/134220702 - (448090660357*x^3)/8858566332 + (443693927*x^4)/44740234 + (2 
5238764095*x^5)/246071287 + (16109860350*x^6)/246071287 + 458281924174/166 
098118725)/((23*x)/225 - (101*x^2)/1350 - (409*x^3)/540 - (457*x^4)/675 + 
(343*x^5)/300 + (11*x^6)/5 + x^7 + 2/75)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.94 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^3} \, dx=\frac {324352970316305 x -1568989318337700 x^{2}+10271022410250000 \,\mathrm {log}\left (5 x +3\right ) x +7013098387669500 x^{5}-68026916543250000 \,\mathrm {log}\left (5 x +3\right ) x^{4}-76102321844062500 \,\mathrm {log}\left (5 x +3\right ) x^{3}-1134028800 \,\mathrm {log}\left (2 x -1\right ) x -7517197561125000 \,\mathrm {log}\left (5 x +3\right ) x^{2}-11093760000 \,\mathrm {log}\left (2 x -1\right ) x^{7}-2869320618020850 x^{3}-12683865600 \,\mathrm {log}\left (2 x -1\right ) x^{5}[phone] \,\mathrm {log}\left (2 x -1\right ) x^{2}[phone] \,\mathrm {log}\left (2 x -1\right ) x^{4}-295833600 \,\mathrm {log}\left (2 x -1\right )+3076101372042300 x^{4}+68026909032363600 \,\mathrm {log}\left (3 x +2\right ) x^{4}+7517196731147400 \,\mathrm {log}\left (3 x +2\right ) x^{2}-3044763606150000 x^{7}+100477393143750000 \,\mathrm {log}\left (5 x +3\right ) x^{7}-100477382049990000 \,\mathrm {log}\left (3 x +2\right ) x^{7}+114879152827687500 \,\mathrm {log}\left (5 x +3\right ) x^{5}-114879140143821900 \,\mathrm {log}\left (3 x +2\right ) x^{5}+76102313441566500 \,\mathrm {log}\left (3 x +2\right ) x^{3}[phone] \,\mathrm {log}\left (2 x -1\right ) x^{3}-24406272000 \,\mathrm {log}\left (2 x -1\right ) x^{6}+221050264916250000 \,\mathrm {log}\left (5 x +3\right ) x^{6}-221050240509978000 \,\mathrm {log}\left (3 x +2\right ) x^{6}+2679397150500000 \,\mathrm {log}\left (5 x +3\right )-2679396854666400 \,\mathrm {log}\left (3 x +2\right )-10271021276221200 \,\mathrm {log}\left (3 x +2\right ) x [phone]}{102316441134600 x^{7}[phone] x^{6}[phone] x^{5}-69272020145944 x^{4}-77495230414910 x^{3}-7654785595996 x^{2}[phone] x [phone]} \] Input:

int(1/(1-2*x)^3/(2+3*x)^4/(3+5*x)^3,x)
 

Output:

(100477393143750000*log(5*x + 3)*x**7 + 221050264916250000*log(5*x + 3)*x* 
*6 + 114879152827687500*log(5*x + 3)*x**5 - 68026916543250000*log(5*x + 3) 
*x**4 - 76102321844062500*log(5*x + 3)*x**3 - 7517197561125000*log(5*x + 3 
)*x**2 + 10271022410250000*log(5*x + 3)*x + 2679397150500000*log(5*x + 3) 
- 100477382049990000*log(3*x + 2)*x**7 - 221050240509978000*log(3*x + 2)*x 
**6 - 114879140143821900*log(3*x + 2)*x**5 + 68026909032363600*log(3*x + 2 
)*x**4 + 76102313441566500*log(3*x + 2)*x**3 + 7517196731147400*log(3*x + 
2)*x**2 - 10271021276221200*log(3*x + 2)*x - 2679396854666400*log(3*x + 2) 
 - 11093760000*log(2*x - 1)*x**7 - 24406272000*log(2*x - 1)*x**6 - 1268386 
5600*log(2*x - 1)*x**5 + 7510886400*log(2*x - 1)*x**4 + 8402496000*log(2*x 
 - 1)*x**3 + 829977600*log(2*x - 1)*x**2 - 1134028800*log(2*x - 1)*x - 295 
833600*log(2*x - 1) - 3044763606150000*x**7 + 7013098387669500*x**5 + 3076 
101372042300*x**4 - 2869320618020850*x**3 - 1568989318337700*x**2 + 324352 
970316305*x + 201107969127184)/(37894978198*(2700*x**7 + 5940*x**6 + 3087* 
x**5 - 1828*x**4 - 2045*x**3 - 202*x**2 + 276*x + 72))