Integrand size = 25, antiderivative size = 105 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^3} \, dx=\frac {11856}{125 (3+2 x)^2}+\frac {35886}{625 (3+2 x)}-\frac {3 (37+47 x)}{10 (3+2 x)^2 \left (2+5 x+3 x^2\right )^2}+\frac {8999+10254 x}{50 (3+2 x)^2 \left (2+5 x+3 x^2\right )}-141 \log (1+x)+\frac {68592 \log (3+2 x)}{3125}+\frac {372033 \log (2+3 x)}{3125} \]
11856/125/(3+2*x)^2+35886/625/(3+2*x)-3/10*(37+47*x)/(3+2*x)^2/(3*x^2+5*x+ 2)^2+1/50*(8999+10254*x)/(3+2*x)^2/(3*x^2+5*x+2)-141*ln(1+x)+68592/3125*ln (3+2*x)+372033/3125*ln(2+3*x)
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^3} \, dx=\frac {-\frac {2600}{(3+2 x)^2}-\frac {24560}{3+2 x}-\frac {75 (653+903 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac {550495+611970 x}{4+10 x+6 x^2}+372033 \log (-4-6 x)-440625 \log (-2 (1+x))+68592 \log (3+2 x)}{3125} \]
(-2600/(3 + 2*x)^2 - 24560/(3 + 2*x) - (75*(653 + 903*x))/(2*(2 + 5*x + 3* x^2)^2) + (550495 + 611970*x)/(4 + 10*x + 6*x^2) + 372033*Log[-4 - 6*x] - 440625*Log[-2*(1 + x)] + 68592*Log[3 + 2*x])/3125
Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89, 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.
\(\displaystyle \int \frac {5-x}{(2 x+3)^3 \left (3 x^2+5 x+2\right )^3} \, dx\) |
\(\Big \downarrow \) 1207 |
\(\displaystyle 27 \int \left (\frac {45728}{28125 (2 x+3)}+\frac {41337}{3125 (3 x+2)}+\frac {9824}{16875 (2 x+3)^2}-\frac {3258}{625 (3 x+2)^2}+\frac {416}{3375 (2 x+3)^3}+\frac {153}{125 (3 x+2)^3}-\frac {47}{9 (x+1)}-\frac {17}{27 (x+1)^2}-\frac {2}{9 (x+1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 27 \left (\frac {17}{27 (x+1)}-\frac {4912}{16875 (2 x+3)}+\frac {1086}{625 (3 x+2)}+\frac {1}{9 (x+1)^2}-\frac {104}{3375 (2 x+3)^2}-\frac {51}{250 (3 x+2)^2}-\frac {47}{9} \log (x+1)+\frac {22864 \log (2 x+3)}{28125}+\frac {13779 \log (3 x+2)}{3125}\right )\) |
27*(1/(9*(1 + x)^2) + 17/(27*(1 + x)) - 104/(3375*(3 + 2*x)^2) - 4912/(168 75*(3 + 2*x)) - 51/(250*(2 + 3*x)^2) + 1086/(625*(2 + 3*x)) - (47*Log[1 + x])/9 + (22864*Log[3 + 2*x])/28125 + (13779*Log[2 + 3*x])/3125)
3.25.4.3.1 Defintions of rubi rules used
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 = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66
method | result | size |
norman | \(\frac {\frac {645948}{625} x^{5}+\frac {3655602}{625} x^{4}+\frac {4435823}{625} x +\frac {8048229}{625} x^{3}+\frac {17180967}{1250} x^{2}+\frac {1771579}{1250}}{\left (3+2 x \right )^{2} \left (3 x^{2}+5 x +2\right )^{2}}+\frac {68592 \ln \left (3+2 x \right )}{3125}-141 \ln \left (1+x \right )+\frac {372033 \ln \left (2+3 x \right )}{3125}\) | \(69\) |
risch | \(\frac {\frac {645948}{625} x^{5}+\frac {3655602}{625} x^{4}+\frac {4435823}{625} x +\frac {8048229}{625} x^{3}+\frac {17180967}{1250} x^{2}+\frac {1771579}{1250}}{\left (3+2 x \right )^{2} \left (3 x^{2}+5 x +2\right )^{2}}+\frac {68592 \ln \left (3+2 x \right )}{3125}-141 \ln \left (1+x \right )+\frac {372033 \ln \left (2+3 x \right )}{3125}\) | \(70\) |
default | \(-\frac {1377}{250 \left (2+3 x \right )^{2}}+\frac {29322}{625 \left (2+3 x \right )}+\frac {372033 \ln \left (2+3 x \right )}{3125}-\frac {104}{125 \left (3+2 x \right )^{2}}-\frac {4912}{625 \left (3+2 x \right )}+\frac {68592 \ln \left (3+2 x \right )}{3125}+\frac {3}{\left (1+x \right )^{2}}+\frac {17}{1+x}-141 \ln \left (1+x \right )\) | \(74\) |
parallelrisch | \(-\frac {-159442110-798448140 x +571050000 \ln \left (1+x \right )-1546287030 x^{2}-116270640 x^{5}-658008360 x^{4}-1448681220 x^{3}-88895232 \ln \left (x +\frac {3}{2}\right )-482154768 \ln \left (x +\frac {2}{3}\right )-482154768 \ln \left (x +\frac {2}{3}\right ) x^{6}+571050000 \ln \left (1+x \right ) x^{6}-88895232 \ln \left (x +\frac {3}{2}\right ) x^{6}+9343012500 \ln \left (1+x \right ) x^{2}-1454424768 \ln \left (x +\frac {3}{2}\right ) x^{2}-7888587732 \ln \left (x +\frac {2}{3}\right ) x^{2}+3616650000 \ln \left (1+x \right ) x^{5}-563003136 \ln \left (x +\frac {3}{2}\right ) x^{5}-3053646864 \ln \left (x +\frac {2}{3}\right ) x^{5}+12594825000 \ln \left (1+x \right ) x^{3}-1960633728 \ln \left (x +\frac {3}{2}\right ) x^{3}-10634191272 \ln \left (x +\frac {2}{3}\right ) x^{3}+9343012500 \ln \left (1+x \right ) x^{4}-1454424768 \ln \left (x +\frac {3}{2}\right ) x^{4}-7888587732 \ln \left (x +\frac {2}{3}\right ) x^{4}+3616650000 \ln \left (1+x \right ) x -563003136 \ln \left (x +\frac {3}{2}\right ) x -3053646864 \ln \left (x +\frac {2}{3}\right ) x}{112500 \left (3+2 x \right )^{2} \left (3 x^{2}+5 x +2\right )^{2}}\) | \(221\) |
(645948/625*x^5+3655602/625*x^4+4435823/625*x+8048229/625*x^3+17180967/125 0*x^2+1771579/1250)/(3+2*x)^2/(3*x^2+5*x+2)^2+68592/3125*ln(3+2*x)-141*ln( 1+x)+372033/3125*ln(2+3*x)
Time = 0.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.63 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^3} \, dx=\frac {6459480 \, x^{5} + 36556020 \, x^{4} + 80482290 \, x^{3} + 85904835 \, x^{2} + 744066 \, {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (3 \, x + 2\right ) + 137184 \, {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (2 \, x + 3\right ) - 881250 \, {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (x + 1\right ) + 44358230 \, x + 8857895}{6250 \, {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )}} \]
1/6250*(6459480*x^5 + 36556020*x^4 + 80482290*x^3 + 85904835*x^2 + 744066* (36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)*log(3*x + 2) + 137184*(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)*lo g(2*x + 3) - 881250*(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228* x + 36)*log(x + 1) + 44358230*x + 8857895)/(36*x^6 + 228*x^5 + 589*x^4 + 7 94*x^3 + 589*x^2 + 228*x + 36)
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^3} \, dx=- \frac {- 1291896 x^{5} - 7311204 x^{4} - 16096458 x^{3} - 17180967 x^{2} - 8871646 x - 1771579}{45000 x^{6} + 285000 x^{5} + 736250 x^{4} + 992500 x^{3} + 736250 x^{2} + 285000 x + 45000} + \frac {372033 \log {\left (x + \frac {2}{3} \right )}}{3125} - 141 \log {\left (x + 1 \right )} + \frac {68592 \log {\left (x + \frac {3}{2} \right )}}{3125} \]
-(-1291896*x**5 - 7311204*x**4 - 16096458*x**3 - 17180967*x**2 - 8871646*x - 1771579)/(45000*x**6 + 285000*x**5 + 736250*x**4 + 992500*x**3 + 736250 *x**2 + 285000*x + 45000) + 372033*log(x + 2/3)/3125 - 141*log(x + 1) + 68 592*log(x + 3/2)/3125
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.78 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^3} \, dx=\frac {1291896 \, x^{5} + 7311204 \, x^{4} + 16096458 \, x^{3} + 17180967 \, x^{2} + 8871646 \, x + 1771579}{1250 \, {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )}} + \frac {372033}{3125} \, \log \left (3 \, x + 2\right ) + \frac {68592}{3125} \, \log \left (2 \, x + 3\right ) - 141 \, \log \left (x + 1\right ) \]
1/1250*(1291896*x^5 + 7311204*x^4 + 16096458*x^3 + 17180967*x^2 + 8871646* x + 1771579)/(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36) + 372033/3125*log(3*x + 2) + 68592/3125*log(2*x + 3) - 141*log(x + 1)
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^3} \, dx=\frac {1291896 \, x^{5} + 7311204 \, x^{4} + 16096458 \, x^{3} + 17180967 \, x^{2} + 8871646 \, x + 1771579}{1250 \, {\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )}^{2}} + \frac {372033}{3125} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {68592}{3125} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 141 \, \log \left ({\left | x + 1 \right |}\right ) \]
1/1250*(1291896*x^5 + 7311204*x^4 + 16096458*x^3 + 17180967*x^2 + 8871646* x + 1771579)/(6*x^3 + 19*x^2 + 19*x + 6)^2 + 372033/3125*log(abs(3*x + 2)) + 68592/3125*log(abs(2*x + 3)) - 141*log(abs(x + 1))
Time = 11.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^3} \, dx=\frac {372033\,\ln \left (x+\frac {2}{3}\right )}{3125}-141\,\ln \left (x+1\right )+\frac {68592\,\ln \left (x+\frac {3}{2}\right )}{3125}+\frac {\frac {17943\,x^5}{625}+\frac {203089\,x^4}{1250}+\frac {2682743\,x^3}{7500}+\frac {5726989\,x^2}{15000}+\frac {4435823\,x}{22500}+\frac {1771579}{45000}}{x^6+\frac {19\,x^5}{3}+\frac {589\,x^4}{36}+\frac {397\,x^3}{18}+\frac {589\,x^2}{36}+\frac {19\,x}{3}+1} \]