Integrand size = 22, antiderivative size = 97 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {343}{18 (2+3 x)^6}+\frac {1617}{5 (2+3 x)^5}+\frac {3927}{(2+3 x)^4}+\frac {42878}{(2+3 x)^3}+\frac {953535}{2 (2+3 x)^2}+\frac {6618975}{2+3 x}-\frac {166375}{2 (3+5 x)^2}+\frac {3584625}{3+5 x}-43848750 \log (2+3 x)+43848750 \log (3+5 x) \]
343/18/(2+3*x)^6+1617/5/(2+3*x)^5+3927/(2+3*x)^4+42878/(2+3*x)^3+953535/2/ (2+3*x)^2+6618975/(2+3*x)-166375/2/(3+5*x)^2+3584625/(3+5*x)-43848750*ln(2 +3*x)+43848750*ln(3+5*x)
Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {343}{18 (2+3 x)^6}+\frac {1617}{5 (2+3 x)^5}+\frac {3927}{(2+3 x)^4}+\frac {42878}{(2+3 x)^3}+\frac {953535}{2 (2+3 x)^2}+\frac {6618975}{2+3 x}-\frac {166375}{2 (3+5 x)^2}+\frac {3584625}{3+5 x}-43848750 \log (5 (2+3 x))+43848750 \log (3+5 x) \]
343/(18*(2 + 3*x)^6) + 1617/(5*(2 + 3*x)^5) + 3927/(2 + 3*x)^4 + 42878/(2 + 3*x)^3 + 953535/(2*(2 + 3*x)^2) + 6618975/(2 + 3*x) - 166375/(2*(3 + 5*x )^2) + 3584625/(3 + 5*x) - 43848750*Log[5*(2 + 3*x)] + 43848750*Log[3 + 5* x]
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.
\(\displaystyle \int \frac {(1-2 x)^3}{(3 x+2)^7 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {219243750}{5 x+3}-\frac {17923125}{(5 x+3)^2}+\frac {831875}{(5 x+3)^3}-\frac {131546250}{3 x+2}-\frac {19856925}{(3 x+2)^2}-\frac {2860605}{(3 x+2)^3}-\frac {385902}{(3 x+2)^4}-\frac {47124}{(3 x+2)^5}-\frac {4851}{(3 x+2)^6}-\frac {343}{(3 x+2)^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6618975}{3 x+2}+\frac {3584625}{5 x+3}+\frac {953535}{2 (3 x+2)^2}-\frac {166375}{2 (5 x+3)^2}+\frac {42878}{(3 x+2)^3}+\frac {3927}{(3 x+2)^4}+\frac {1617}{5 (3 x+2)^5}+\frac {343}{18 (3 x+2)^6}-43848750 \log (3 x+2)+43848750 \log (5 x+3)\) |
343/(18*(2 + 3*x)^6) + 1617/(5*(2 + 3*x)^5) + 3927/(2 + 3*x)^4 + 42878/(2 + 3*x)^3 + 953535/(2*(2 + 3*x)^2) + 6618975/(2 + 3*x) - 166375/(2*(3 + 5*x )^2) + 3584625/(3 + 5*x) - 43848750*Log[2 + 3*x] + 43848750*Log[3 + 5*x]
3.15.32.3.1 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 = 0.81 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70
method | result | size |
norman | \(\frac {53276231250 x^{7}+243294789375 x^{6}+337112732979 x^{3}+476013260250 x^{5}+\frac {429157211632}{15} x +\frac {1034482340925}{2} x^{4}+\frac {2372106406093}{18} x^{2}+\frac {26610779007}{10}}{\left (2+3 x \right )^{6} \left (3+5 x \right )^{2}}-43848750 \ln \left (2+3 x \right )+43848750 \ln \left (3+5 x \right )\) | \(68\) |
risch | \(\frac {53276231250 x^{7}+243294789375 x^{6}+337112732979 x^{3}+476013260250 x^{5}+\frac {429157211632}{15} x +\frac {1034482340925}{2} x^{4}+\frac {2372106406093}{18} x^{2}+\frac {26610779007}{10}}{\left (2+3 x \right )^{6} \left (3+5 x \right )^{2}}-43848750 \ln \left (2+3 x \right )+43848750 \ln \left (3+5 x \right )\) | \(69\) |
default | \(\frac {343}{18 \left (2+3 x \right )^{6}}+\frac {1617}{5 \left (2+3 x \right )^{5}}+\frac {3927}{\left (2+3 x \right )^{4}}+\frac {42878}{\left (2+3 x \right )^{3}}+\frac {953535}{2 \left (2+3 x \right )^{2}}+\frac {6618975}{2+3 x}-\frac {166375}{2 \left (3+5 x \right )^{2}}+\frac {3584625}{3+5 x}-43848750 \ln \left (2+3 x \right )+43848750 \ln \left (3+5 x \right )\) | \(90\) |
parallelrisch | \(-\frac {24246604799040 x -9678436416000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+29823323904000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-1794248755200000 \ln \left (x +\frac {3}{5}\right ) x +9678436416000000 \ln \left (\frac {2}{3}+x \right ) x^{2}+1794248755200000 \ln \left (\frac {2}{3}+x \right ) x +4710352617196308 x^{5}+4334335931147787 x^{6}+2215032434493390 x^{7}+1200431443187520 x^{3}+3070385357587380 x^{4}+260651001598480 x^{2}+484981447402575 x^{8}+57418990992000000 \ln \left (\frac {2}{3}+x \right ) x^{4}+145479628800000 \ln \left (\frac {2}{3}+x \right )+23935945176000000 \ln \left (\frac {2}{3}+x \right ) x^{7}-23935945176000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-145479628800000 \ln \left (x +\frac {3}{5}\right )+70730377027200000 \ln \left (\frac {2}{3}+x \right ) x^{5}-29823323904000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-70730377027200000 \ln \left (x +\frac {3}{5}\right ) x^{5}-57418990992000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+54438931720800000 \ln \left (\frac {2}{3}+x \right ) x^{6}-54438931720800000 \ln \left (x +\frac {3}{5}\right ) x^{6}+4603066380000000 \ln \left (\frac {2}{3}+x \right ) x^{8}-4603066380000000 \ln \left (x +\frac {3}{5}\right ) x^{8}}{5760 \left (2+3 x \right )^{6} \left (3+5 x \right )^{2}}\) | \(208\) |
(53276231250*x^7+243294789375*x^6+337112732979*x^3+476013260250*x^5+429157 211632/15*x+1034482340925/2*x^4+2372106406093/18*x^2+26610779007/10)/(2+3* x)^6/(3+5*x)^2-43848750*ln(2+3*x)+43848750*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.80 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {4794860812500 \, x^{7} + 21896531043750 \, x^{6} + 42841193422500 \, x^{5} + 46551705341625 \, x^{4} + 30340145968110 \, x^{3} + 11860532030465 \, x^{2} + 3946387500 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )} \log \left (5 \, x + 3\right ) - 3946387500 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )} \log \left (3 \, x + 2\right ) + 2574943269792 \, x + 239497011063}{90 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )}} \]
1/90*(4794860812500*x^7 + 21896531043750*x^6 + 42841193422500*x^5 + 465517 05341625*x^4 + 30340145968110*x^3 + 11860532030465*x^2 + 3946387500*(18225 *x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 383 20*x^2 + 7104*x + 576)*log(5*x + 3) - 3946387500*(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 5 76)*log(3*x + 2) + 2574943269792*x + 239497011063)/(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576)
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^3} \, dx=- \frac {- 4794860812500 x^{7} - 21896531043750 x^{6} - 42841193422500 x^{5} - 46551705341625 x^{4} - 30340145968110 x^{3} - 11860532030465 x^{2} - 2574943269792 x - 239497011063}{1640250 x^{8} + 8529300 x^{7} + 19398690 x^{6} + 25203960 x^{5} + 20460600 x^{4} + 10627200 x^{3} + 3448800 x^{2} + 639360 x + 51840} + 43848750 \log {\left (x + \frac {3}{5} \right )} - 43848750 \log {\left (x + \frac {2}{3} \right )} \]
-(-4794860812500*x**7 - 21896531043750*x**6 - 42841193422500*x**5 - 465517 05341625*x**4 - 30340145968110*x**3 - 11860532030465*x**2 - 2574943269792* x - 239497011063)/(1640250*x**8 + 8529300*x**7 + 19398690*x**6 + 25203960* x**5 + 20460600*x**4 + 10627200*x**3 + 3448800*x**2 + 639360*x + 51840) + 43848750*log(x + 3/5) - 43848750*log(x + 2/3)
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {4794860812500 \, x^{7} + 21896531043750 \, x^{6} + 42841193422500 \, x^{5} + 46551705341625 \, x^{4} + 30340145968110 \, x^{3} + 11860532030465 \, x^{2} + 2574943269792 \, x + 239497011063}{90 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )}} + 43848750 \, \log \left (5 \, x + 3\right ) - 43848750 \, \log \left (3 \, x + 2\right ) \]
1/90*(4794860812500*x^7 + 21896531043750*x^6 + 42841193422500*x^5 + 465517 05341625*x^4 + 30340145968110*x^3 + 11860532030465*x^2 + 2574943269792*x + 239497011063)/(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x ^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576) + 43848750*log(5*x + 3) - 4384 8750*log(3*x + 2)
Time = 0.37 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {4794860812500 \, x^{7} + 21896531043750 \, x^{6} + 42841193422500 \, x^{5} + 46551705341625 \, x^{4} + 30340145968110 \, x^{3} + 11860532030465 \, x^{2} + 2574943269792 \, x + 239497011063}{90 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{6}} + 43848750 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 43848750 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]
1/90*(4794860812500*x^7 + 21896531043750*x^6 + 42841193422500*x^5 + 465517 05341625*x^4 + 30340145968110*x^3 + 11860532030465*x^2 + 2574943269792*x + 239497011063)/((5*x + 3)^2*(3*x + 2)^6) + 43848750*log(abs(5*x + 3)) - 43 848750*log(abs(3*x + 2))
Time = 1.38 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {2923250\,x^7+\frac {40048525\,x^6}{3}+\frac {705204830\,x^5}{27}+\frac {170285159\,x^4}{6}+\frac {12485656777\,x^3}{675}+\frac {2372106406093\,x^2}{328050}+\frac {429157211632\,x}{273375}+\frac {2956753223}{20250}}{x^8+\frac {26\,x^7}{5}+\frac {887\,x^6}{75}+\frac {10372\,x^5}{675}+\frac {1684\,x^4}{135}+\frac {2624\,x^3}{405}+\frac {7664\,x^2}{3645}+\frac {2368\,x}{6075}+\frac {64}{2025}}-87697500\,\mathrm {atanh}\left (30\,x+19\right ) \]
((429157211632*x)/273375 + (2372106406093*x^2)/328050 + (12485656777*x^3)/ 675 + (170285159*x^4)/6 + (705204830*x^5)/27 + (40048525*x^6)/3 + 2923250* x^7 + 2956753223/20250)/((2368*x)/6075 + (7664*x^2)/3645 + (2624*x^3)/405 + (1684*x^4)/135 + (10372*x^5)/675 + (887*x^6)/75 + (26*x^7)/5 + x^8 + 64/ 2025) - 87697500*atanh(30*x + 19)