Integrand size = 22, antiderivative size = 73 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^3} \, dx=\frac {424432 x}{390625}-\frac {62097 x^2}{31250}-\frac {393 x^3}{625}+\frac {22977 x^4}{6250}-\frac {324 x^5}{3125}-\frac {324 x^6}{125}-\frac {1331}{3906250 (3+5 x)^2}-\frac {19239}{1953125 (3+5 x)}+\frac {109032 \log (3+5 x)}{1953125} \] Output:
424432/390625*x-62097/31250*x^2-393/625*x^3+22977/6250*x^4-324/3125*x^5-32 4/125*x^6-1331/3906250/(3+5*x)^2-19239/(5859375+9765625*x)+109032/1953125* ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^3} \, dx=\frac {151973789+698557830 x+711123525 x^2-744310000 x^3-692475000 x^4+1828837500 x^5+1278703125 x^6-1569375000 x^7-1265625000 x^8+1090320 (3+5 x)^2 \log (6 (3+5 x))}{19531250 (3+5 x)^2} \] Input:
Integrate[((1 - 2*x)^3*(2 + 3*x)^5)/(3 + 5*x)^3,x]
Output:
(151973789 + 698557830*x + 711123525*x^2 - 744310000*x^3 - 692475000*x^4 + 1828837500*x^5 + 1278703125*x^6 - 1569375000*x^7 - 1265625000*x^8 + 10903 20*(3 + 5*x)^2*Log[6*(3 + 5*x)])/(19531250*(3 + 5*x)^2)
Time = 0.20 (sec) , antiderivative size = 73, 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.
| \(\displaystyle \int \frac {(1-2 x)^3 (3 x+2)^5}{(5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {1944 x^5}{125}-\frac {324 x^4}{625}+\frac {45954 x^3}{3125}-\frac {1179 x^2}{625}-\frac {62097 x}{15625}+\frac {109032}{390625 (5 x+3)}+\frac {19239}{390625 (5 x+3)^2}+\frac {1331}{390625 (5 x+3)^3}+\frac {424432}{390625}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {324 x^6}{125}-\frac {324 x^5}{3125}+\frac {22977 x^4}{6250}-\frac {393 x^3}{625}-\frac {62097 x^2}{31250}+\frac {424432 x}{390625}-\frac {19239}{1953125 (5 x+3)}-\frac {1331}{3906250 (5 x+3)^2}+\frac {109032 \log (5 x+3)}{1953125}\) |
Input:
Int[((1 - 2*x)^3*(2 + 3*x)^5)/(3 + 5*x)^3,x]
Output:
(424432*x)/390625 - (62097*x^2)/31250 - (393*x^3)/625 + (22977*x^4)/6250 - (324*x^5)/3125 - (324*x^6)/125 - 1331/(3906250*(3 + 5*x)^2) - 19239/(1953 125*(3 + 5*x)) + (109032*Log[3 + 5*x])/1953125
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 = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {324 x^{6}}{125}-\frac {324 x^{5}}{3125}+\frac {22977 x^{4}}{6250}-\frac {393 x^{3}}{625}-\frac {62097 x^{2}}{31250}+\frac {424432 x}{390625}+\frac {-\frac {19239 x}{390625}-\frac {23353}{781250}}{\left (3+5 x \right )^{2}}+\frac {109032 \ln \left (3+5 x \right )}{1953125}\) | \(52\) |
| default | \(-\frac {324 x^{6}}{125}-\frac {324 x^{5}}{3125}+\frac {22977 x^{4}}{6250}-\frac {393 x^{3}}{625}-\frac {62097 x^{2}}{31250}+\frac {424432 x}{390625}-\frac {19239}{1953125 \left (3+5 x \right )}+\frac {109032 \ln \left (3+5 x \right )}{1953125}-\frac {1331}{3906250 \left (3+5 x \right )^{2}}\) | \(56\) |
| norman | \(\frac {\frac {11518712}{1171875} x +\frac {10403068}{703125} x^{2}-\frac {595448}{15625} x^{3}-\frac {110796}{3125} x^{4}+\frac {292614}{3125} x^{5}+\frac {81837}{1250} x^{6}-\frac {10044}{125} x^{7}-\frac {324}{5} x^{8}}{\left (3+5 x \right )^{2}}+\frac {109032 \ln \left (3+5 x \right )}{1953125}\) | \(57\) |
| parallelrisch | \(\frac {-2278125000 x^{8}-2824875000 x^{7}+2301665625 x^{6}+3291907500 x^{5}-1246455000 x^{4}+49064400 \ln \left (x +\frac {3}{5}\right ) x^{2}-1339758000 x^{3}+58877280 \ln \left (x +\frac {3}{5}\right ) x +520153400 x^{2}+17663184 \ln \left (x +\frac {3}{5}\right )+345561360 x}{35156250 \left (3+5 x \right )^{2}}\) | \(71\) |
| meijerg | \(\frac {16 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {8 x^{2}}{9 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {56 x \left (15 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {109032 \ln \left (1+\frac {5 x}{3}\right )}{1953125}-\frac {154 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {63 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {7749 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{6250 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {729 x \left (-\frac {21875}{243} x^{5}+\frac {8750}{81} x^{4}-\frac {4375}{27} x^{3}+\frac {3500}{9} x^{2}+1050 x +420\right )}{31250 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {72171 x \left (\frac {125000}{729} x^{6}-\frac {43750}{243} x^{5}+\frac {17500}{81} x^{4}-\frac {8750}{27} x^{3}+\frac {7000}{9} x^{2}+2100 x +840\right )}{781250 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {26244 x \left (-\frac {390625}{729} x^{7}+\frac {125000}{243} x^{6}-\frac {43750}{81} x^{5}+\frac {17500}{27} x^{4}-\frac {8750}{9} x^{3}+\frac {7000}{3} x^{2}+6300 x +2520\right )}{1953125 \left (1+\frac {5 x}{3}\right )^{2}}\) | \(247\) |
Input:
int((1-2*x)^3*(2+3*x)^5/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
-324/125*x^6-324/3125*x^5+22977/6250*x^4-393/625*x^3-62097/31250*x^2+42443 2/390625*x+25*(-19239/9765625*x-23353/19531250)/(3+5*x)^2+109032/1953125*l n(3+5*x)
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^3} \, dx=-\frac {253125000 \, x^{8} + 313875000 \, x^{7} - 255740625 \, x^{6} - 365767500 \, x^{5} + 138495000 \, x^{4} + 148862000 \, x^{3} - 57470475 \, x^{2} - 218064 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 38006490 \, x + 116765}{3906250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:
integrate((1-2*x)^3*(2+3*x)^5/(3+5*x)^3,x, algorithm="fricas")
Output:
-1/3906250*(253125000*x^8 + 313875000*x^7 - 255740625*x^6 - 365767500*x^5 + 138495000*x^4 + 148862000*x^3 - 57470475*x^2 - 218064*(25*x^2 + 30*x + 9 )*log(5*x + 3) - 38006490*x + 116765)/(25*x^2 + 30*x + 9)
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^3} \, dx=- \frac {324 x^{6}}{125} - \frac {324 x^{5}}{3125} + \frac {22977 x^{4}}{6250} - \frac {393 x^{3}}{625} - \frac {62097 x^{2}}{31250} + \frac {424432 x}{390625} - \frac {38478 x + 23353}{19531250 x^{2} + 23437500 x + 7031250} + \frac {109032 \log {\left (5 x + 3 \right )}}{1953125} \] Input:
integrate((1-2*x)**3*(2+3*x)**5/(3+5*x)**3,x)
Output:
-324*x**6/125 - 324*x**5/3125 + 22977*x**4/6250 - 393*x**3/625 - 62097*x** 2/31250 + 424432*x/390625 - (38478*x + 23353)/(19531250*x**2 + 23437500*x + 7031250) + 109032*log(5*x + 3)/1953125
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^3} \, dx=-\frac {324}{125} \, x^{6} - \frac {324}{3125} \, x^{5} + \frac {22977}{6250} \, x^{4} - \frac {393}{625} \, x^{3} - \frac {62097}{31250} \, x^{2} + \frac {424432}{390625} \, x - \frac {121 \, {\left (318 \, x + 193\right )}}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {109032}{1953125} \, \log \left (5 \, x + 3\right ) \] Input:
integrate((1-2*x)^3*(2+3*x)^5/(3+5*x)^3,x, algorithm="maxima")
Output:
-324/125*x^6 - 324/3125*x^5 + 22977/6250*x^4 - 393/625*x^3 - 62097/31250*x ^2 + 424432/390625*x - 121/781250*(318*x + 193)/(25*x^2 + 30*x + 9) + 1090 32/1953125*log(5*x + 3)
Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^3} \, dx=-\frac {324}{125} \, x^{6} - \frac {324}{3125} \, x^{5} + \frac {22977}{6250} \, x^{4} - \frac {393}{625} \, x^{3} - \frac {62097}{31250} \, x^{2} + \frac {424432}{390625} \, x - \frac {121 \, {\left (318 \, x + 193\right )}}{781250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {109032}{1953125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \] Input:
integrate((1-2*x)^3*(2+3*x)^5/(3+5*x)^3,x, algorithm="giac")
Output:
-324/125*x^6 - 324/3125*x^5 + 22977/6250*x^4 - 393/625*x^3 - 62097/31250*x ^2 + 424432/390625*x - 121/781250*(318*x + 193)/(5*x + 3)^2 + 109032/19531 25*log(abs(5*x + 3))
Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^3} \, dx=\frac {424432\,x}{390625}+\frac {109032\,\ln \left (x+\frac {3}{5}\right )}{1953125}-\frac {\frac {19239\,x}{9765625}+\frac {23353}{19531250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {62097\,x^2}{31250}-\frac {393\,x^3}{625}+\frac {22977\,x^4}{6250}-\frac {324\,x^5}{3125}-\frac {324\,x^6}{125} \] Input:
int(-((2*x - 1)^3*(3*x + 2)^5)/(5*x + 3)^3,x)
Output:
(424432*x)/390625 + (109032*log(x + 3/5))/1953125 - ((19239*x)/9765625 + 2 3353/19531250)/((6*x)/5 + x^2 + 9/25) - (62097*x^2)/31250 - (393*x^3)/625 + (22977*x^4)/6250 - (324*x^5)/3125 - (324*x^6)/125
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^3} \, dx=\frac {5451600 \,\mathrm {log}\left (5 x +3\right ) x^{2}+6541920 \,\mathrm {log}\left (5 x +3\right ) x +1962576 \,\mathrm {log}\left (5 x +3\right )-253125000 x^{8}-313875000 x^{7}+255740625 x^{6}+365767500 x^{5}-138495000 x^{4}-148862000 x^{3}+25798400 x^{2}-11518712}{97656250 x^{2}+117187500 x +35156250} \] Input:
int((1-2*x)^3*(2+3*x)^5/(3+5*x)^3,x)
Output:
(5451600*log(5*x + 3)*x**2 + 6541920*log(5*x + 3)*x + 1962576*log(5*x + 3) - 253125000*x**8 - 313875000*x**7 + 255740625*x**6 + 365767500*x**5 - 138 495000*x**4 - 148862000*x**3 + 25798400*x**2 - 11518712)/(3906250*(25*x**2 + 30*x + 9))