Integrand size = 22, antiderivative size = 66 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {117649}{7744 (1-2 x)^2}-\frac {67228}{1331 (1-2 x)}-\frac {31347 x}{2000}-\frac {729 x^2}{400}-\frac {1}{831875 (3+5 x)}-\frac {7383075 \log (1-2 x)}{234256}+\frac {204 \log (3+5 x)}{9150625} \] Output:
117649/7744/(1-2*x)^2-67228/(1331-2662*x)-31347/2000*x-729/400*x^2-1/(2495 625+4159375*x)-7383075/234256*ln(1-2*x)+204/9150625*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.12 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {117649}{7744 (1-2 x)^2}+\frac {2187}{250} (1-2 x)-\frac {729 (1-2 x)^2}{1600}+\frac {67228}{1331 (-1+2 x)}-\frac {1}{831875 (3+5 x)}-\frac {7383075 \log (1-2 x)}{234256}+\frac {204 \log (6+10 x)}{9150625} \] Input:
Integrate[(2 + 3*x)^6/((1 - 2*x)^3*(3 + 5*x)^2),x]
Output:
117649/(7744*(1 - 2*x)^2) + (2187*(1 - 2*x))/250 - (729*(1 - 2*x)^2)/1600 + 67228/(1331*(-1 + 2*x)) - 1/(831875*(3 + 5*x)) - (7383075*Log[1 - 2*x])/ 234256 + (204*Log[6 + 10*x])/9150625
Time = 0.23 (sec) , antiderivative size = 66, 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.
| \(\displaystyle \int \frac {(3 x+2)^6}{(1-2 x)^3 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {729 x}{200}-\frac {7383075}{117128 (2 x-1)}+\frac {204}{1830125 (5 x+3)}-\frac {134456}{1331 (2 x-1)^2}+\frac {1}{166375 (5 x+3)^2}-\frac {117649}{1936 (2 x-1)^3}-\frac {31347}{2000}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {729 x^2}{400}-\frac {31347 x}{2000}-\frac {67228}{1331 (1-2 x)}-\frac {1}{831875 (5 x+3)}+\frac {117649}{7744 (1-2 x)^2}-\frac {7383075 \log (1-2 x)}{234256}+\frac {204 \log (5 x+3)}{9150625}\) |
Input:
Int[(2 + 3*x)^6/((1 - 2*x)^3*(3 + 5*x)^2),x]
Output:
117649/(7744*(1 - 2*x)^2) - 67228/(1331*(1 - 2*x)) - (31347*x)/2000 - (729 *x^2)/400 - 1/(831875*(3 + 5*x)) - (7383075*Log[1 - 2*x])/234256 + (204*Lo g[3 + 5*x])/9150625
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 = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.79
| method | result | size |
| risch | \(-\frac {729 x^{2}}{400}-\frac {31347 x}{2000}+\frac {\frac {420174996}{831875} x^{2}+\frac {6733304631}{53240000} x -\frac {5640849439}{53240000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {7383075 \ln \left (-1+2 x \right )}{234256}+\frac {204 \ln \left (3+5 x \right )}{9150625}\) | \(52\) |
| default | \(-\frac {729 x^{2}}{400}-\frac {31347 x}{2000}-\frac {1}{831875 \left (3+5 x \right )}+\frac {204 \ln \left (3+5 x \right )}{9150625}+\frac {117649}{7744 \left (-1+2 x \right )^{2}}+\frac {67228}{1331 \left (-1+2 x \right )}-\frac {7383075 \ln \left (-1+2 x \right )}{234256}\) | \(53\) |
| norman | \(\frac {\frac {326232091}{998250} x^{2}-\frac {669903661}{3993000} x +\frac {3372039701}{3993000} x^{3}-\frac {29889}{100} x^{4}-\frac {729}{20} x^{5}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {7383075 \ln \left (-1+2 x \right )}{234256}+\frac {204 \ln \left (3+5 x \right )}{9150625}\) | \(57\) |
| parallelrisch | \(-\frac {16009933500 x^{5}+276865312500 \ln \left (x -\frac {1}{2}\right ) x^{3}-195840 \ln \left (x +\frac {3}{5}\right ) x^{3}+131281454700 x^{4}-110746125000 \ln \left (x -\frac {1}{2}\right ) x^{2}+78336 \ln \left (x +\frac {3}{5}\right ) x^{2}-370924367110 x^{3}-96902859375 \ln \left (x -\frac {1}{2}\right ) x +68544 \ln \left (x +\frac {3}{5}\right ) x -143542120040 x^{2}+41529796875 \ln \left (x -\frac {1}{2}\right )-29376 \ln \left (x +\frac {3}{5}\right )+73689402710 x}{439230000 \left (-1+2 x \right )^{2} \left (3+5 x \right )}\) | \(103\) |
Input:
int((2+3*x)^6/(1-2*x)^3/(3+5*x)^2,x,method=_RETURNVERBOSE)
Output:
-729/400*x^2-31347/2000*x+20*(105043749/4159375*x^2+6733304631/1064800000* x-5640849439/1064800000)/(-1+2*x)^2/(3+5*x)-7383075/234256*ln(-1+2*x)+204/ 9150625*ln(3+5*x)
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.36 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {21346578000 \, x^{5} + 175041939600 \, x^{4} - 80903530620 \, x^{3} - 356854410264 \, x^{2} - 13056 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 18457687500 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 46529265321 \, x + 62049343829}{585640000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \] Input:
integrate((2+3*x)^6/(1-2*x)^3/(3+5*x)^2,x, algorithm="fricas")
Output:
-1/585640000*(21346578000*x^5 + 175041939600*x^4 - 80903530620*x^3 - 35685 4410264*x^2 - 13056*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) + 18457687500* (20*x^3 - 8*x^2 - 7*x + 3)*log(2*x - 1) - 46529265321*x + 62049343829)/(20 *x^3 - 8*x^2 - 7*x + 3)
Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^2} \, dx=- \frac {729 x^{2}}{400} - \frac {31347 x}{2000} - \frac {- 26891199744 x^{2} - 6733304631 x + 5640849439}{1064800000 x^{3} - 425920000 x^{2} - 372680000 x + 159720000} - \frac {7383075 \log {\left (x - \frac {1}{2} \right )}}{234256} + \frac {204 \log {\left (x + \frac {3}{5} \right )}}{9150625} \] Input:
integrate((2+3*x)**6/(1-2*x)**3/(3+5*x)**2,x)
Output:
-729*x**2/400 - 31347*x/2000 - (-26891199744*x**2 - 6733304631*x + 5640849 439)/(1064800000*x**3 - 425920000*x**2 - 372680000*x + 159720000) - 738307 5*log(x - 1/2)/234256 + 204*log(x + 3/5)/9150625
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {729}{400} \, x^{2} - \frac {31347}{2000} \, x + \frac {26891199744 \, x^{2} + 6733304631 \, x - 5640849439}{53240000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac {204}{9150625} \, \log \left (5 \, x + 3\right ) - \frac {7383075}{234256} \, \log \left (2 \, x - 1\right ) \] Input:
integrate((2+3*x)^6/(1-2*x)^3/(3+5*x)^2,x, algorithm="maxima")
Output:
-729/400*x^2 - 31347/2000*x + 1/53240000*(26891199744*x^2 + 6733304631*x - 5640849439)/(20*x^3 - 8*x^2 - 7*x + 3) + 204/9150625*log(5*x + 3) - 73830 75/234256*log(2*x - 1)
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.42 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {{\left (5 \, x + 3\right )}^{2} {\left (\frac {555011028}{5 \, x + 3} - \frac {13845990449}{{\left (5 \, x + 3\right )}^{2}} + \frac {50757096489}{{\left (5 \, x + 3\right )}^{3}} + 21346578\right )}}{73205000 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2}} - \frac {1}{831875 \, {\left (5 \, x + 3\right )}} + \frac {315171}{10000} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {7383075}{234256} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \] Input:
integrate((2+3*x)^6/(1-2*x)^3/(3+5*x)^2,x, algorithm="giac")
Output:
-1/73205000*(5*x + 3)^2*(555011028/(5*x + 3) - 13845990449/(5*x + 3)^2 + 5 0757096489/(5*x + 3)^3 + 21346578)/(11/(5*x + 3) - 2)^2 - 1/831875/(5*x + 3) + 315171/10000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 7383075/234256*log(a bs(-11/(5*x + 3) + 2))
Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {204\,\ln \left (x+\frac {3}{5}\right )}{9150625}-\frac {7383075\,\ln \left (x-\frac {1}{2}\right )}{234256}-\frac {31347\,x}{2000}-\frac {\frac {105043749\,x^2}{4159375}+\frac {6733304631\,x}{1064800000}-\frac {5640849439}{1064800000}}{-x^3+\frac {2\,x^2}{5}+\frac {7\,x}{20}-\frac {3}{20}}-\frac {729\,x^2}{400} \] Input:
int(-(3*x + 2)^6/((2*x - 1)^3*(5*x + 3)^2),x)
Output:
(204*log(x + 3/5))/9150625 - (7383075*log(x - 1/2))/234256 - (31347*x)/200 0 - ((6733304631*x)/1064800000 + (105043749*x^2)/4159375 - 5640849439/1064 800000)/((7*x)/20 + (2*x^2)/5 - x^3 - 3/20) - (729*x^2)/400
Time = 0.15 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.76 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {65280 \,\mathrm {log}\left (5 x +3\right ) x^{3}-26112 \,\mathrm {log}\left (5 x +3\right ) x^{2}-22848 \,\mathrm {log}\left (5 x +3\right ) x +9792 \,\mathrm {log}\left (5 x +3\right )-92288437500 \,\mathrm {log}\left (2 x -1\right ) x^{3}+36915375000 \,\mathrm {log}\left (2 x -1\right ) x^{2}+32300953125 \,\mathrm {log}\left (2 x -1\right ) x -13843265625 \,\mathrm {log}\left (2 x -1\right )-5336644500 x^{5}-43760484900 x^{4}+243259889070 x^{3}-66429585915 x +17942765005}{2928200000 x^{3}-1171280000 x^{2}-1024870000 x +439230000} \] Input:
int((2+3*x)^6/(1-2*x)^3/(3+5*x)^2,x)
Output:
(65280*log(5*x + 3)*x**3 - 26112*log(5*x + 3)*x**2 - 22848*log(5*x + 3)*x + 9792*log(5*x + 3) - 92288437500*log(2*x - 1)*x**3 + 36915375000*log(2*x - 1)*x**2 + 32300953125*log(2*x - 1)*x - 13843265625*log(2*x - 1) - 533664 4500*x**5 - 43760484900*x**4 + 243259889070*x**3 - 66429585915*x + 1794276 5005)/(146410000*(20*x**3 - 8*x**2 - 7*x + 3))