Integrand size = 22, antiderivative size = 66 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {49}{3645 (2+3 x)^5}-\frac {763}{2916 (2+3 x)^4}+\frac {4099}{2187 (2+3 x)^3}-\frac {8285}{1458 (2+3 x)^2}+\frac {3800}{729 (2+3 x)}+\frac {500}{729} \log (2+3 x) \] Output:
49/3645/(2+3*x)^5-763/2916/(2+3*x)^4+4099/2187/(2+3*x)^3-8285/1458/(2+3*x) ^2+3800/(1458+2187*x)+500/729*ln(2+3*x)
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {1965218+13889625 x+36564120 x^2+42537150 x^3+18468000 x^4+30000 (2+3 x)^5 \log (20+30 x)}{43740 (2+3 x)^5} \] Input:
Integrate[((1 - 2*x)^2*(3 + 5*x)^3)/(2 + 3*x)^6,x]
Output:
(1965218 + 13889625*x + 36564120*x^2 + 42537150*x^3 + 18468000*x^4 + 30000 *(2 + 3*x)^5*Log[20 + 30*x])/(43740*(2 + 3*x)^5)
Time = 0.24 (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.
\(\displaystyle \int \frac {(1-2 x)^2 (5 x+3)^3}{(3 x+2)^6} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {500}{243 (3 x+2)}-\frac {3800}{243 (3 x+2)^2}+\frac {8285}{243 (3 x+2)^3}-\frac {4099}{243 (3 x+2)^4}+\frac {763}{243 (3 x+2)^5}-\frac {49}{243 (3 x+2)^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3800}{729 (3 x+2)}-\frac {8285}{1458 (3 x+2)^2}+\frac {4099}{2187 (3 x+2)^3}-\frac {763}{2916 (3 x+2)^4}+\frac {49}{3645 (3 x+2)^5}+\frac {500}{729} \log (3 x+2)\) |
Input:
Int[((1 - 2*x)^2*(3 + 5*x)^3)/(2 + 3*x)^6,x]
Output:
49/(3645*(2 + 3*x)^5) - 763/(2916*(2 + 3*x)^4) + 4099/(2187*(2 + 3*x)^3) - 8285/(1458*(2 + 3*x)^2) + 3800/(729*(2 + 3*x)) + (500*Log[2 + 3*x])/729
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 = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58
method | result | size |
norman | \(\frac {\frac {1945}{2} x^{3}+\frac {3800}{9} x^{4}+\frac {203134}{243} x^{2}+\frac {925975}{2916} x +\frac {982609}{21870}}{\left (2+3 x \right )^{5}}+\frac {500 \ln \left (2+3 x \right )}{729}\) | \(38\) |
risch | \(\frac {\frac {1945}{2} x^{3}+\frac {3800}{9} x^{4}+\frac {203134}{243} x^{2}+\frac {925975}{2916} x +\frac {982609}{21870}}{\left (2+3 x \right )^{5}}+\frac {500 \ln \left (2+3 x \right )}{729}\) | \(39\) |
default | \(-\frac {763}{2916 \left (2+3 x \right )^{4}}+\frac {500 \ln \left (2+3 x \right )}{729}-\frac {8285}{1458 \left (2+3 x \right )^{2}}+\frac {3800}{729 \left (2+3 x \right )}+\frac {49}{3645 \left (2+3 x \right )^{5}}+\frac {4099}{2187 \left (2+3 x \right )^{3}}\) | \(55\) |
parallelrisch | \(\frac {38880000 \ln \left (\frac {2}{3}+x \right ) x^{5}+129600000 \ln \left (\frac {2}{3}+x \right ) x^{4}-79591329 x^{5}+172800000 \ln \left (\frac {2}{3}+x \right ) x^{3}-166808430 x^{4}+115200000 \ln \left (\frac {2}{3}+x \right ) x^{2}-126874440 x^{3}+38400000 \ln \left (\frac {2}{3}+x \right ) x -40817520 x^{2}+5120000 \ln \left (\frac {2}{3}+x \right )-4530720 x}{233280 \left (2+3 x \right )^{5}}\) | \(83\) |
meijerg | \(\frac {27 x \left (\frac {81}{16} x^{4}+\frac {135}{8} x^{3}+\frac {45}{2} x^{2}+15 x +5\right )}{320 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {27 x^{2} \left (\frac {27}{8} x^{3}+\frac {45}{4} x^{2}+15 x +10\right )}{1280 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {69 x^{3} \left (\frac {9}{4} x^{2}+\frac {15}{2} x +10\right )}{640 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {47 x^{4} \left (\frac {3 x}{2}+5\right )}{256 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {5 x^{5}}{4 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {25 x \left (\frac {11097}{16} x^{4}+\frac {10395}{8} x^{3}+\frac {2115}{2} x^{2}+405 x +60\right )}{1458 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {500 \ln \left (1+\frac {3 x}{2}\right )}{729}\) | \(148\) |
Input:
int((1-2*x)^2*(3+5*x)^3/(2+3*x)^6,x,method=_RETURNVERBOSE)
Output:
(1945/2*x^3+3800/9*x^4+203134/243*x^2+925975/2916*x+982609/21870)/(2+3*x)^ 5+500/729*ln(2+3*x)
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.24 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {18468000 \, x^{4} + 42537150 \, x^{3} + 36564120 \, x^{2} + 30000 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 13889625 \, x + 1965218}{43740 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \] Input:
integrate((1-2*x)^2*(3+5*x)^3/(2+3*x)^6,x, algorithm="fricas")
Output:
1/43740*(18468000*x^4 + 42537150*x^3 + 36564120*x^2 + 30000*(243*x^5 + 810 *x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(3*x + 2) + 13889625*x + 196521 8)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {18468000 x^{4} + 42537150 x^{3} + 36564120 x^{2} + 13889625 x + 1965218}{10628820 x^{5} + 35429400 x^{4} + 47239200 x^{3} + 31492800 x^{2} + 10497600 x + 1399680} + \frac {500 \log {\left (3 x + 2 \right )}}{729} \] Input:
integrate((1-2*x)**2*(3+5*x)**3/(2+3*x)**6,x)
Output:
(18468000*x**4 + 42537150*x**3 + 36564120*x**2 + 13889625*x + 1965218)/(10 628820*x**5 + 35429400*x**4 + 47239200*x**3 + 31492800*x**2 + 10497600*x + 1399680) + 500*log(3*x + 2)/729
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {18468000 \, x^{4} + 42537150 \, x^{3} + 36564120 \, x^{2} + 13889625 \, x + 1965218}{43740 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {500}{729} \, \log \left (3 \, x + 2\right ) \] Input:
integrate((1-2*x)^2*(3+5*x)^3/(2+3*x)^6,x, algorithm="maxima")
Output:
1/43740*(18468000*x^4 + 42537150*x^3 + 36564120*x^2 + 13889625*x + 1965218 )/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 500/729*log(3*x + 2)
Time = 0.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.59 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {18468000 \, x^{4} + 42537150 \, x^{3} + 36564120 \, x^{2} + 13889625 \, x + 1965218}{43740 \, {\left (3 \, x + 2\right )}^{5}} + \frac {500}{729} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \] Input:
integrate((1-2*x)^2*(3+5*x)^3/(2+3*x)^6,x, algorithm="giac")
Output:
1/43740*(18468000*x^4 + 42537150*x^3 + 36564120*x^2 + 13889625*x + 1965218 )/(3*x + 2)^5 + 500/729*log(abs(3*x + 2))
Time = 1.00 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {500\,\ln \left (x+\frac {2}{3}\right )}{729}+\frac {\frac {3800\,x^4}{2187}+\frac {1945\,x^3}{486}+\frac {203134\,x^2}{59049}+\frac {925975\,x}{708588}+\frac {982609}{5314410}}{x^5+\frac {10\,x^4}{3}+\frac {40\,x^3}{9}+\frac {80\,x^2}{27}+\frac {80\,x}{81}+\frac {32}{243}} \] Input:
int(((2*x - 1)^2*(5*x + 3)^3)/(3*x + 2)^6,x)
Output:
(500*log(x + 2/3))/729 + ((925975*x)/708588 + (203134*x^2)/59049 + (1945*x ^3)/486 + (3800*x^4)/2187 + 982609/5314410)/((80*x)/81 + (80*x^2)/27 + (40 *x^3)/9 + (10*x^4)/3 + x^5 + 32/243)
Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.65 \[ \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {7290000 \,\mathrm {log}\left (3 x +2\right ) x^{5}+24300000 \,\mathrm {log}\left (3 x +2\right ) x^{4}+32400000 \,\mathrm {log}\left (3 x +2\right ) x^{3}+21600000 \,\mathrm {log}\left (3 x +2\right ) x^{2}+7200000 \,\mathrm {log}\left (3 x +2\right ) x +960000 \,\mathrm {log}\left (3 x +2\right )-5540400 x^{5}+17913150 x^{3}+20148120 x^{2}+8417625 x +1235618}{10628820 x^{5}+35429400 x^{4}+47239200 x^{3}+31492800 x^{2}+10497600 x +1399680} \] Input:
int((1-2*x)^2*(3+5*x)^3/(2+3*x)^6,x)
Output:
(7290000*log(3*x + 2)*x**5 + 24300000*log(3*x + 2)*x**4 + 32400000*log(3*x + 2)*x**3 + 21600000*log(3*x + 2)*x**2 + 7200000*log(3*x + 2)*x + 960000* log(3*x + 2) - 5540400*x**5 + 17913150*x**3 + 20148120*x**2 + 8417625*x + 1235618)/(43740*(243*x**5 + 810*x**4 + 1080*x**3 + 720*x**2 + 240*x + 32))