Integrand size = 22, antiderivative size = 66 \[ \int \frac {(2+3 x)^6}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {117649}{21296 (1-2 x)}+\frac {2916 x}{625}+\frac {729 x^2}{1000}-\frac {1}{756250 (3+5 x)^2}-\frac {202}{4159375 (3+5 x)}+\frac {1563051 \log (1-2 x)}{234256}+\frac {17139 \log (3+5 x)}{45753125} \] Output:
117649/(21296-42592*x)+2916/625*x+729/1000*x^2-1/756250/(3+5*x)^2-202/(124 78125+20796875*x)+1563051/234256*ln(1-2*x)+17139/45753125*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^6}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {11 \left (-334753155+\frac {735306250}{1-2 x}+620991360 x+97029900 x^2-\frac {176}{(3+5 x)^2}-\frac {6464}{3+5 x}\right )+9769068750 \log (1-2 x)+548448 \log (6+10 x)}{1464100000} \] Input:
Integrate[(2 + 3*x)^6/((1 - 2*x)^2*(3 + 5*x)^3),x]
Output:
(11*(-334753155 + 735306250/(1 - 2*x) + 620991360*x + 97029900*x^2 - 176/( 3 + 5*x)^2 - 6464/(3 + 5*x)) + 9769068750*Log[1 - 2*x] + 548448*Log[6 + 10 *x])/1464100000
Time = 0.22 (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.
\(\displaystyle \int \frac {(3 x+2)^6}{(1-2 x)^2 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {729 x}{500}+\frac {1563051}{117128 (2 x-1)}+\frac {17139}{9150625 (5 x+3)}+\frac {117649}{10648 (2 x-1)^2}+\frac {202}{831875 (5 x+3)^2}+\frac {1}{75625 (5 x+3)^3}+\frac {2916}{625}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {729 x^2}{1000}+\frac {2916 x}{625}+\frac {117649}{21296 (1-2 x)}-\frac {202}{4159375 (5 x+3)}-\frac {1}{756250 (5 x+3)^2}+\frac {1563051 \log (1-2 x)}{234256}+\frac {17139 \log (5 x+3)}{45753125}\) |
Input:
Int[(2 + 3*x)^6/((1 - 2*x)^2*(3 + 5*x)^3),x]
Output:
117649/(21296*(1 - 2*x)) + (2916*x)/625 + (729*x^2)/1000 - 1/(756250*(3 + 5*x)^2) - 202/(4159375*(3 + 5*x)) + (1563051*Log[1 - 2*x])/234256 + (17139 *Log[3 + 5*x])/45753125
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 | \(\frac {729 x^{2}}{1000}+\frac {2916 x}{625}+\frac {-\frac {1838272089}{13310000} x^{2}-\frac {5514798579}{33275000} x -\frac {3308868341}{66550000}}{\left (-1+2 x \right ) \left (3+5 x \right )^{2}}+\frac {1563051 \ln \left (-1+2 x \right )}{234256}+\frac {17139 \ln \left (3+5 x \right )}{45753125}\) | \(52\) |
default | \(\frac {729 x^{2}}{1000}+\frac {2916 x}{625}-\frac {1}{756250 \left (3+5 x \right )^{2}}-\frac {202}{4159375 \left (3+5 x \right )}+\frac {17139 \ln \left (3+5 x \right )}{45753125}-\frac {117649}{21296 \left (-1+2 x \right )}+\frac {1563051 \ln \left (-1+2 x \right )}{234256}\) | \(53\) |
norman | \(\frac {-\frac {23599588033}{59895000} x^{2}-\frac {1457537849}{11979000} x^{3}-\frac {2823670147}{19965000} x +\frac {51759}{200} x^{4}+\frac {729}{20} x^{5}}{\left (-1+2 x \right ) \left (3+5 x \right )^{2}}+\frac {1563051 \ln \left (-1+2 x \right )}{234256}+\frac {17139 \ln \left (3+5 x \right )}{45753125}\) | \(57\) |
parallelrisch | \(\frac {240149002500 x^{5}+2198040468750 \ln \left (x -\frac {1}{2}\right ) x^{3}+123400800 \ln \left (x +\frac {3}{5}\right ) x^{3}+1705057917750 x^{4}+1538628328125 \ln \left (x -\frac {1}{2}\right ) x^{2}+86380560 \ln \left (x +\frac {3}{5}\right ) x^{2}-801645816950 x^{3}-527529712500 \ln \left (x -\frac {1}{2}\right ) x -29616192 \ln \left (x +\frac {3}{5}\right ) x -2595954683630 x^{2}-395647284375 \ln \left (x -\frac {1}{2}\right )-22212144 \ln \left (x +\frac {3}{5}\right )-931811148510 x}{6588450000 \left (-1+2 x \right ) \left (3+5 x \right )^{2}}\) | \(103\) |
Input:
int((2+3*x)^6/(1-2*x)^2/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
729/1000*x^2+2916/625*x+50*(-1838272089/665500000*x^2-5514798579/166375000 0*x-3308868341/3327500000)/(-1+2*x)/(3+5*x)^2+1563051/234256*ln(-1+2*x)+17 139/45753125*ln(3+5*x)
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.36 \[ \int \frac {(2+3 x)^6}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {26683222500 \, x^{5} + 189450879750 \, x^{4} + 113136863400 \, x^{3} - 146893374705 \, x^{2} + 274224 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 4884534375 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) - 152064641058 \, x - 36397551751}{732050000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \] Input:
integrate((2+3*x)^6/(1-2*x)^2/(3+5*x)^3,x, algorithm="fricas")
Output:
1/732050000*(26683222500*x^5 + 189450879750*x^4 + 113136863400*x^3 - 14689 3374705*x^2 + 274224*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) + 488453437 5*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x - 1) - 152064641058*x - 36397551751 )/(50*x^3 + 35*x^2 - 12*x - 9)
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^6}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {729 x^{2}}{1000} + \frac {2916 x}{625} + \frac {- 9191360445 x^{2} - 11029597158 x - 3308868341}{3327500000 x^{3} + 2329250000 x^{2} - 798600000 x - 598950000} + \frac {1563051 \log {\left (x - \frac {1}{2} \right )}}{234256} + \frac {17139 \log {\left (x + \frac {3}{5} \right )}}{45753125} \] Input:
integrate((2+3*x)**6/(1-2*x)**2/(3+5*x)**3,x)
Output:
729*x**2/1000 + 2916*x/625 + (-9191360445*x**2 - 11029597158*x - 330886834 1)/(3327500000*x**3 + 2329250000*x**2 - 798600000*x - 598950000) + 1563051 *log(x - 1/2)/234256 + 17139*log(x + 3/5)/45753125
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \frac {(2+3 x)^6}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {729}{1000} \, x^{2} + \frac {2916}{625} \, x - \frac {9191360445 \, x^{2} + 11029597158 \, x + 3308868341}{66550000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac {17139}{45753125} \, \log \left (5 \, x + 3\right ) + \frac {1563051}{234256} \, \log \left (2 \, x - 1\right ) \] Input:
integrate((2+3*x)^6/(1-2*x)^2/(3+5*x)^3,x, algorithm="maxima")
Output:
729/1000*x^2 + 2916/625*x - 1/66550000*(9191360445*x^2 + 11029597158*x + 3 308868341)/(50*x^3 + 35*x^2 - 12*x - 9) + 17139/45753125*log(5*x + 3) + 15 63051/234256*log(2*x - 1)
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.42 \[ \int \frac {(2+3 x)^6}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {{\left (2 \, x - 1\right )}^{2} {\left (\frac {25615893600}{2 \, x - 1} + \frac {93337977265}{{\left (2 \, x - 1\right )}^{2}} + \frac {95568773322}{{\left (2 \, x - 1\right )}^{3}} + 1334161125\right )}}{292820000 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2}} - \frac {117649}{21296 \, {\left (2 \, x - 1\right )}} - \frac {333639}{50000} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {17139}{45753125} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \] Input:
integrate((2+3*x)^6/(1-2*x)^2/(3+5*x)^3,x, algorithm="giac")
Output:
1/292820000*(2*x - 1)^2*(25615893600/(2*x - 1) + 93337977265/(2*x - 1)^2 + 95568773322/(2*x - 1)^3 + 1334161125)/(11/(2*x - 1) + 5)^2 - 117649/21296 /(2*x - 1) - 333639/50000*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 17139/457531 25*log(abs(-11/(2*x - 1) - 5))
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^6}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {2916\,x}{625}+\frac {1563051\,\ln \left (x-\frac {1}{2}\right )}{234256}+\frac {17139\,\ln \left (x+\frac {3}{5}\right )}{45753125}+\frac {\frac {1838272089\,x^2}{665500000}+\frac {5514798579\,x}{1663750000}+\frac {3308868341}{3327500000}}{-x^3-\frac {7\,x^2}{10}+\frac {6\,x}{25}+\frac {9}{50}}+\frac {729\,x^2}{1000} \] Input:
int((3*x + 2)^6/((2*x - 1)^2*(5*x + 3)^3),x)
Output:
(2916*x)/625 + (1563051*log(x - 1/2))/234256 + (17139*log(x + 3/5))/457531 25 + ((5514798579*x)/1663750000 + (1838272089*x^2)/665500000 + 3308868341/ 3327500000)/((6*x)/25 - (7*x^2)/10 - x^3 + 9/50) + (729*x^2)/1000
Time = 0.15 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.76 \[ \int \frac {(2+3 x)^6}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {13711200 \,\mathrm {log}\left (5 x +3\right ) x^{3}+9597840 \,\mathrm {log}\left (5 x +3\right ) x^{2}-3290688 \,\mathrm {log}\left (5 x +3\right ) x -2468016 \,\mathrm {log}\left (5 x +3\right )+244226718750 \,\mathrm {log}\left (2 x -1\right ) x^{3}+170958703125 \,\mathrm {log}\left (2 x -1\right ) x^{2}-58614412500 \,\mathrm {log}\left (2 x -1\right ) x -43960809375 \,\mathrm {log}\left (2 x -1\right )+26683222500 x^{5}+189450879750 x^{4}+322984541550 x^{3}-202428083814 x -74170133818}{36602500000 x^{3}+25621750000 x^{2}-8784600000 x -6588450000} \] Input:
int((2+3*x)^6/(1-2*x)^2/(3+5*x)^3,x)
Output:
(13711200*log(5*x + 3)*x**3 + 9597840*log(5*x + 3)*x**2 - 3290688*log(5*x + 3)*x - 2468016*log(5*x + 3) + 244226718750*log(2*x - 1)*x**3 + 170958703 125*log(2*x - 1)*x**2 - 58614412500*log(2*x - 1)*x - 43960809375*log(2*x - 1) + 26683222500*x**5 + 189450879750*x**4 + 322984541550*x**3 - 202428083 814*x - 74170133818)/(732050000*(50*x**3 + 35*x**2 - 12*x - 9))