Integrand size = 22, antiderivative size = 72 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{3+5 x} \, dx=\frac {41666223 x}{1953125}+\frac {11111259 x^2}{781250}-\frac {17453753 x^3}{234375}-\frac {5848749 x^4}{62500}+\frac {2212083 x^5}{15625}+\frac {331713 x^6}{1250}-\frac {40338 x^7}{875}-\frac {13851 x^8}{50}-\frac {648 x^9}{5}+\frac {1331 \log (3+5 x)}{9765625} \]
41666223/1953125*x+11111259/781250*x^2-17453753/234375*x^3-5848749/62500*x ^4+2212083/15625*x^5+331713/1250*x^6-40338/875*x^7-13851/50*x^8-648/5*x^9+ 1331/9765625*ln(3+5*x)
Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{3+5 x} \, dx=\frac {18072649071+87499068300 x+58334109750 x^2-305440677500 x^3-383824153125 x^4+580671787500 x^5+1088433281250 x^6-189084375000 x^7-1136214843750 x^8-531562500000 x^9+559020 \log (3+5 x)}{4101562500} \]
(18072649071 + 87499068300*x + 58334109750*x^2 - 305440677500*x^3 - 383824 153125*x^4 + 580671787500*x^5 + 1088433281250*x^6 - 189084375000*x^7 - 113 6214843750*x^8 - 531562500000*x^9 + 559020*Log[3 + 5*x])/4101562500
Time = 0.19 (sec) , antiderivative size = 72, 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.
\(\displaystyle \int \frac {(1-2 x)^3 (3 x+2)^6}{5 x+3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {5832 x^8}{5}-\frac {55404 x^7}{25}-\frac {40338 x^6}{125}+\frac {995139 x^5}{625}+\frac {2212083 x^4}{3125}-\frac {5848749 x^3}{15625}-\frac {17453753 x^2}{78125}+\frac {11111259 x}{390625}+\frac {1331}{1953125 (5 x+3)}+\frac {41666223}{1953125}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {648 x^9}{5}-\frac {13851 x^8}{50}-\frac {40338 x^7}{875}+\frac {331713 x^6}{1250}+\frac {2212083 x^5}{15625}-\frac {5848749 x^4}{62500}-\frac {17453753 x^3}{234375}+\frac {11111259 x^2}{781250}+\frac {41666223 x}{1953125}+\frac {1331 \log (5 x+3)}{9765625}\) |
(41666223*x)/1953125 + (11111259*x^2)/781250 - (17453753*x^3)/234375 - (58 48749*x^4)/62500 + (2212083*x^5)/15625 + (331713*x^6)/1250 - (40338*x^7)/8 75 - (13851*x^8)/50 - (648*x^9)/5 + (1331*Log[3 + 5*x])/9765625
3.14.88.3.1 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 = 2.45 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71
method | result | size |
parallelrisch | \(-\frac {648 x^{9}}{5}-\frac {13851 x^{8}}{50}-\frac {40338 x^{7}}{875}+\frac {331713 x^{6}}{1250}+\frac {2212083 x^{5}}{15625}-\frac {5848749 x^{4}}{62500}-\frac {17453753 x^{3}}{234375}+\frac {11111259 x^{2}}{781250}+\frac {41666223 x}{1953125}+\frac {1331 \ln \left (x +\frac {3}{5}\right )}{9765625}\) | \(51\) |
default | \(\frac {41666223 x}{1953125}+\frac {11111259 x^{2}}{781250}-\frac {17453753 x^{3}}{234375}-\frac {5848749 x^{4}}{62500}+\frac {2212083 x^{5}}{15625}+\frac {331713 x^{6}}{1250}-\frac {40338 x^{7}}{875}-\frac {13851 x^{8}}{50}-\frac {648 x^{9}}{5}+\frac {1331 \ln \left (3+5 x \right )}{9765625}\) | \(53\) |
norman | \(\frac {41666223 x}{1953125}+\frac {11111259 x^{2}}{781250}-\frac {17453753 x^{3}}{234375}-\frac {5848749 x^{4}}{62500}+\frac {2212083 x^{5}}{15625}+\frac {331713 x^{6}}{1250}-\frac {40338 x^{7}}{875}-\frac {13851 x^{8}}{50}-\frac {648 x^{9}}{5}+\frac {1331 \ln \left (3+5 x \right )}{9765625}\) | \(53\) |
risch | \(\frac {41666223 x}{1953125}+\frac {11111259 x^{2}}{781250}-\frac {17453753 x^{3}}{234375}-\frac {5848749 x^{4}}{62500}+\frac {2212083 x^{5}}{15625}+\frac {331713 x^{6}}{1250}-\frac {40338 x^{7}}{875}-\frac {13851 x^{8}}{50}-\frac {648 x^{9}}{5}+\frac {1331 \ln \left (3+5 x \right )}{9765625}\) | \(53\) |
meijerg | \(\frac {1331 \ln \left (1+\frac {5 x}{3}\right )}{9765625}+\frac {192 x}{5}+\frac {177147 x \left (-\frac {2734375}{243} x^{7}+\frac {625000}{81} x^{6}-\frac {437500}{81} x^{5}+\frac {35000}{9} x^{4}-\frac {8750}{3} x^{3}+\frac {7000}{3} x^{2}-2100 x +2520\right )}{5468750}-\frac {531441 x \left (\frac {109375000}{6561} x^{8}-\frac {2734375}{243} x^{7}+\frac {625000}{81} x^{6}-\frac {437500}{81} x^{5}+\frac {35000}{9} x^{4}-\frac {8750}{3} x^{3}+\frac {7000}{3} x^{2}-2100 x +2520\right )}{68359375}-\frac {336 x \left (\frac {100}{9} x^{2}-10 x +12\right )}{25}+\frac {264 x \left (-5 x +6\right )}{25}-\frac {80919 x \left (-\frac {218750}{243} x^{5}+\frac {17500}{27} x^{4}-\frac {4375}{9} x^{3}+\frac {3500}{9} x^{2}-350 x +420\right )}{312500}+\frac {56133 x \left (\frac {2500}{27} x^{4}-\frac {625}{9} x^{3}+\frac {500}{9} x^{2}-50 x +60\right )}{15625}-\frac {567 x \left (-\frac {625}{9} x^{3}+\frac {500}{9} x^{2}-50 x +60\right )}{3125}-\frac {1003833 x \left (\frac {625000}{243} x^{6}-\frac {437500}{243} x^{5}+\frac {35000}{27} x^{4}-\frac {8750}{9} x^{3}+\frac {7000}{9} x^{2}-700 x +840\right )}{10937500}\) | \(217\) |
-648/5*x^9-13851/50*x^8-40338/875*x^7+331713/1250*x^6+2212083/15625*x^5-58 48749/62500*x^4-17453753/234375*x^3+11111259/781250*x^2+41666223/1953125*x +1331/9765625*ln(x+3/5)
Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{3+5 x} \, dx=-\frac {648}{5} \, x^{9} - \frac {13851}{50} \, x^{8} - \frac {40338}{875} \, x^{7} + \frac {331713}{1250} \, x^{6} + \frac {2212083}{15625} \, x^{5} - \frac {5848749}{62500} \, x^{4} - \frac {17453753}{234375} \, x^{3} + \frac {11111259}{781250} \, x^{2} + \frac {41666223}{1953125} \, x + \frac {1331}{9765625} \, \log \left (5 \, x + 3\right ) \]
-648/5*x^9 - 13851/50*x^8 - 40338/875*x^7 + 331713/1250*x^6 + 2212083/1562 5*x^5 - 5848749/62500*x^4 - 17453753/234375*x^3 + 11111259/781250*x^2 + 41 666223/1953125*x + 1331/9765625*log(5*x + 3)
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{3+5 x} \, dx=- \frac {648 x^{9}}{5} - \frac {13851 x^{8}}{50} - \frac {40338 x^{7}}{875} + \frac {331713 x^{6}}{1250} + \frac {2212083 x^{5}}{15625} - \frac {5848749 x^{4}}{62500} - \frac {17453753 x^{3}}{234375} + \frac {11111259 x^{2}}{781250} + \frac {41666223 x}{1953125} + \frac {1331 \log {\left (5 x + 3 \right )}}{9765625} \]
-648*x**9/5 - 13851*x**8/50 - 40338*x**7/875 + 331713*x**6/1250 + 2212083* x**5/15625 - 5848749*x**4/62500 - 17453753*x**3/234375 + 11111259*x**2/781 250 + 41666223*x/1953125 + 1331*log(5*x + 3)/9765625
Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{3+5 x} \, dx=-\frac {648}{5} \, x^{9} - \frac {13851}{50} \, x^{8} - \frac {40338}{875} \, x^{7} + \frac {331713}{1250} \, x^{6} + \frac {2212083}{15625} \, x^{5} - \frac {5848749}{62500} \, x^{4} - \frac {17453753}{234375} \, x^{3} + \frac {11111259}{781250} \, x^{2} + \frac {41666223}{1953125} \, x + \frac {1331}{9765625} \, \log \left (5 \, x + 3\right ) \]
-648/5*x^9 - 13851/50*x^8 - 40338/875*x^7 + 331713/1250*x^6 + 2212083/1562 5*x^5 - 5848749/62500*x^4 - 17453753/234375*x^3 + 11111259/781250*x^2 + 41 666223/1953125*x + 1331/9765625*log(5*x + 3)
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{3+5 x} \, dx=-\frac {648}{5} \, x^{9} - \frac {13851}{50} \, x^{8} - \frac {40338}{875} \, x^{7} + \frac {331713}{1250} \, x^{6} + \frac {2212083}{15625} \, x^{5} - \frac {5848749}{62500} \, x^{4} - \frac {17453753}{234375} \, x^{3} + \frac {11111259}{781250} \, x^{2} + \frac {41666223}{1953125} \, x + \frac {1331}{9765625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]
-648/5*x^9 - 13851/50*x^8 - 40338/875*x^7 + 331713/1250*x^6 + 2212083/1562 5*x^5 - 5848749/62500*x^4 - 17453753/234375*x^3 + 11111259/781250*x^2 + 41 666223/1953125*x + 1331/9765625*log(abs(5*x + 3))
Time = 0.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{3+5 x} \, dx=\frac {41666223\,x}{1953125}+\frac {1331\,\ln \left (x+\frac {3}{5}\right )}{9765625}+\frac {11111259\,x^2}{781250}-\frac {17453753\,x^3}{234375}-\frac {5848749\,x^4}{62500}+\frac {2212083\,x^5}{15625}+\frac {331713\,x^6}{1250}-\frac {40338\,x^7}{875}-\frac {13851\,x^8}{50}-\frac {648\,x^9}{5} \]