Integrand size = 25, antiderivative size = 67 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^2} \, dx=-\frac {6}{1+x}-\frac {26}{25 (3+2 x)^2}-\frac {812}{125 (3+2 x)}-\frac {459}{125 (2+3 x)}-\log (1+x)+\frac {8104}{625} \log (3+2 x)-\frac {7479}{625} \log (2+3 x) \] Output:
-6/(1+x)-26/25/(3+2*x)^2-812/(375+250*x)-459/(250+375*x)-ln(1+x)+8104/625* ln(3+2*x)-7479/625*ln(2+3*x)
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^2} \, dx=\frac {1}{625} \left (-\frac {650}{(3+2 x)^2}-\frac {4060}{3+2 x}-\frac {15 (653+903 x)}{2+5 x+3 x^2}-7479 \log (-4-6 x)-625 \log (-2 (1+x))+8104 \log (3+2 x)\right ) \] Input:
Integrate[(5 - x)/((3 + 2*x)^3*(2 + 5*x + 3*x^2)^2),x]
Output:
(-650/(3 + 2*x)^2 - 4060/(3 + 2*x) - (15*(653 + 903*x))/(2 + 5*x + 3*x^2) - 7479*Log[-4 - 6*x] - 625*Log[-2*(1 + x)] + 8104*Log[3 + 2*x])/625
Time = 0.40 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1207, 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.
\(\displaystyle \int \frac {5-x}{(2 x+3)^3 \left (3 x^2+5 x+2\right )^2} \, dx\) |
\(\Big \downarrow \) 1207 |
\(\displaystyle 9 \int \left (\frac {16208}{5625 (2 x+3)}-\frac {2493}{625 (3 x+2)}+\frac {1624}{1125 (2 x+3)^2}+\frac {153}{125 (3 x+2)^2}+\frac {104}{225 (2 x+3)^3}-\frac {1}{9 (x+1)}+\frac {2}{3 (x+1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 9 \left (-\frac {2}{3 (x+1)}-\frac {812}{1125 (2 x+3)}-\frac {51}{125 (3 x+2)}-\frac {26}{225 (2 x+3)^2}-\frac {1}{9} \log (x+1)+\frac {8104 \log (2 x+3)}{5625}-\frac {831}{625} \log (3 x+2)\right )\) |
Input:
Int[(5 - x)/((3 + 2*x)^3*(2 + 5*x + 3*x^2)^2),x]
Output:
9*(-2/(3*(1 + x)) - 26/(225*(3 + 2*x)^2) - 812/(1125*(3 + 2*x)) - 51/(125* (2 + 3*x)) - Log[1 + x]/9 + (8104*Log[3 + 2*x])/5625 - (831*Log[2 + 3*x])/ 625)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1 /c^p Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[p, -1] && IntegersQ[m, n] && NiceSqrtQ[b^2 - 4* a*c]
Time = 1.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {459}{125 \left (3 x +2\right )}-\frac {7479 \ln \left (3 x +2\right )}{625}-\frac {6}{x +1}-\ln \left (x +1\right )-\frac {26}{25 \left (2 x +3\right )^{2}}-\frac {812}{125 \left (2 x +3\right )}+\frac {8104 \ln \left (2 x +3\right )}{625}\) | \(58\) |
norman | \(\frac {-\frac {63967}{125} x -\frac {56162}{125} x^{2}-\frac {15708}{125} x^{3}-\frac {22763}{125}}{\left (2 x +3\right )^{2} \left (3 x^{2}+5 x +2\right )}-\ln \left (x +1\right )+\frac {8104 \ln \left (2 x +3\right )}{625}-\frac {7479 \ln \left (3 x +2\right )}{625}\) | \(59\) |
risch | \(\frac {-\frac {63967}{125} x -\frac {56162}{125} x^{2}-\frac {15708}{125} x^{3}-\frac {22763}{125}}{\left (2 x +3\right )^{2} \left (3 x^{2}+5 x +2\right )}-\ln \left (x +1\right )+\frac {8104 \ln \left (2 x +3\right )}{625}-\frac {7479 \ln \left (3 x +2\right )}{625}\) | \(60\) |
parallelrisch | \(-\frac {90000 \ln \left (x +1\right ) x^{4}+1076976 \ln \left (x +\frac {2}{3}\right ) x^{4}-1166976 \ln \left (x +\frac {3}{2}\right ) x^{4}+1365780+420000 \ln \left (x +1\right ) x^{3}+5025888 \ln \left (x +\frac {2}{3}\right ) x^{3}-5445888 \ln \left (x +\frac {3}{2}\right ) x^{3}+712500 \ln \left (x +1\right ) x^{2}+8526060 \ln \left (x +\frac {2}{3}\right ) x^{2}-9238560 \ln \left (x +\frac {3}{2}\right ) x^{2}+942480 x^{3}+517500 \ln \left (x +1\right ) x +6192612 \ln \left (x +\frac {2}{3}\right ) x -6710112 \ln \left (x +\frac {3}{2}\right ) x +3369720 x^{2}+135000 \ln \left (x +1\right )+1615464 \ln \left (x +\frac {2}{3}\right )-1750464 \ln \left (x +\frac {3}{2}\right )+3838020 x}{7500 \left (2 x +3\right )^{2} \left (3 x^{2}+5 x +2\right )}\) | \(157\) |
Input:
int((5-x)/(2*x+3)^3/(3*x^2+5*x+2)^2,x,method=_RETURNVERBOSE)
Output:
-459/125/(3*x+2)-7479/625*ln(3*x+2)-6/(x+1)-ln(x+1)-26/25/(2*x+3)^2-812/12 5/(2*x+3)+8104/625*ln(2*x+3)
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (57) = 114\).
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.81 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^2} \, dx=-\frac {78540 \, x^{3} + 280810 \, x^{2} + 7479 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (3 \, x + 2\right ) - 8104 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (2 \, x + 3\right ) + 625 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (x + 1\right ) + 319835 \, x + 113815}{625 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )}} \] Input:
integrate((5-x)/(3+2*x)^3/(3*x^2+5*x+2)^2,x, algorithm="fricas")
Output:
-1/625*(78540*x^3 + 280810*x^2 + 7479*(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 1 8)*log(3*x + 2) - 8104*(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18)*log(2*x + 3) + 625*(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18)*log(x + 1) + 319835*x + 1138 15)/(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18)
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^2} \, dx=- \frac {15708 x^{3} + 56162 x^{2} + 63967 x + 22763}{1500 x^{4} + 7000 x^{3} + 11875 x^{2} + 8625 x + 2250} - \frac {7479 \log {\left (x + \frac {2}{3} \right )}}{625} - \log {\left (x + 1 \right )} + \frac {8104 \log {\left (x + \frac {3}{2} \right )}}{625} \] Input:
integrate((5-x)/(3+2*x)**3/(3*x**2+5*x+2)**2,x)
Output:
-(15708*x**3 + 56162*x**2 + 63967*x + 22763)/(1500*x**4 + 7000*x**3 + 1187 5*x**2 + 8625*x + 2250) - 7479*log(x + 2/3)/625 - log(x + 1) + 8104*log(x + 3/2)/625
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^2} \, dx=-\frac {15708 \, x^{3} + 56162 \, x^{2} + 63967 \, x + 22763}{125 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )}} - \frac {7479}{625} \, \log \left (3 \, x + 2\right ) + \frac {8104}{625} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) \] Input:
integrate((5-x)/(3+2*x)^3/(3*x^2+5*x+2)^2,x, algorithm="maxima")
Output:
-1/125*(15708*x^3 + 56162*x^2 + 63967*x + 22763)/(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18) - 7479/625*log(3*x + 2) + 8104/625*log(2*x + 3) - log(x + 1)
Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^2} \, dx=-\frac {15708 \, x^{3} + 56162 \, x^{2} + 63967 \, x + 22763}{125 \, {\left (3 \, x + 2\right )} {\left (2 \, x + 3\right )}^{2} {\left (x + 1\right )}} - \frac {7479}{625} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {8104}{625} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \log \left ({\left | x + 1 \right |}\right ) \] Input:
integrate((5-x)/(3+2*x)^3/(3*x^2+5*x+2)^2,x, algorithm="giac")
Output:
-1/125*(15708*x^3 + 56162*x^2 + 63967*x + 22763)/((3*x + 2)*(2*x + 3)^2*(x + 1)) - 7479/625*log(abs(3*x + 2)) + 8104/625*log(abs(2*x + 3)) - log(abs (x + 1))
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^2} \, dx=\frac {8104\,\ln \left (x+\frac {3}{2}\right )}{625}-\frac {7479\,\ln \left (x+\frac {2}{3}\right )}{625}-\ln \left (x+1\right )-\frac {\frac {1309\,x^3}{125}+\frac {28081\,x^2}{750}+\frac {63967\,x}{1500}+\frac {22763}{1500}}{x^4+\frac {14\,x^3}{3}+\frac {95\,x^2}{12}+\frac {23\,x}{4}+\frac {3}{2}} \] Input:
int(-(x - 5)/((2*x + 3)^3*(5*x + 3*x^2 + 2)^2),x)
Output:
(8104*log(x + 3/2))/625 - (7479*log(x + 2/3))/625 - log(x + 1) - ((63967*x )/1500 + (28081*x^2)/750 + (1309*x^3)/125 + 22763/1500)/((23*x)/4 + (95*x^ 2)/12 + (14*x^3)/3 + x^4 + 3/2)
Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.66 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^2} \, dx=\frac {-179496 \,\mathrm {log}\left (3 x +2\right ) x^{4}-837648 \,\mathrm {log}\left (3 x +2\right ) x^{3}-1421010 \,\mathrm {log}\left (3 x +2\right ) x^{2}-1032102 \,\mathrm {log}\left (3 x +2\right ) x -269244 \,\mathrm {log}\left (3 x +2\right )+194496 \,\mathrm {log}\left (2 x +3\right ) x^{4}+907648 \,\mathrm {log}\left (2 x +3\right ) x^{3}+1539760 \,\mathrm {log}\left (2 x +3\right ) x^{2}+1118352 \,\mathrm {log}\left (2 x +3\right ) x +291744 \,\mathrm {log}\left (2 x +3\right )-15000 \,\mathrm {log}\left (x +1\right ) x^{4}-70000 \,\mathrm {log}\left (x +1\right ) x^{3}-118750 \,\mathrm {log}\left (x +1\right ) x^{2}-86250 \,\mathrm {log}\left (x +1\right ) x -22500 \,\mathrm {log}\left (x +1\right )+33660 x^{4}-295145 x^{2}-446125 x -177140}{15000 x^{4}+70000 x^{3}+118750 x^{2}+86250 x +22500} \] Input:
int((5-x)/(3+2*x)^3/(3*x^2+5*x+2)^2,x)
Output:
( - 179496*log(3*x + 2)*x**4 - 837648*log(3*x + 2)*x**3 - 1421010*log(3*x + 2)*x**2 - 1032102*log(3*x + 2)*x - 269244*log(3*x + 2) + 194496*log(2*x + 3)*x**4 + 907648*log(2*x + 3)*x**3 + 1539760*log(2*x + 3)*x**2 + 1118352 *log(2*x + 3)*x + 291744*log(2*x + 3) - 15000*log(x + 1)*x**4 - 70000*log( x + 1)*x**3 - 118750*log(x + 1)*x**2 - 86250*log(x + 1)*x - 22500*log(x + 1) + 33660*x**4 - 295145*x**2 - 446125*x - 177140)/(1250*(12*x**4 + 56*x** 3 + 95*x**2 + 69*x + 18))