Integrand size = 25, antiderivative size = 55 \[ \int \frac {1}{(5-2 x) (4+3 x) \left (2+3 x+x^2\right )^2} \, dx=-\frac {1}{7 (1+x)}+\frac {1}{18 (2+x)}-\frac {16 \log (5-2 x)}{91287}-\frac {33}{49} \log (1+x)-\frac {67}{324} \log (2+x)+\frac {81}{92} \log (4+3 x) \] Output:
-1/7/(1+x)+1/(36+18*x)-16/91287*ln(5-2*x)-33/49*ln(1+x)-67/324*ln(2+x)+81/ 92*ln(4+3*x)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(5-2 x) (4+3 x) \left (2+3 x+x^2\right )^2} \, dx=-\frac {29+11 x}{126 \left (2+3 x+x^2\right )}-\frac {16 \log (5-2 x)}{91287}-\frac {33}{49} \log (2 (1+x))-\frac {67}{324} \log (2 (2+x))+\frac {81}{92} \log (8+6 x) \] Input:
Integrate[1/((5 - 2*x)*(4 + 3*x)*(2 + 3*x + x^2)^2),x]
Output:
-1/126*(29 + 11*x)/(2 + 3*x + x^2) - (16*Log[5 - 2*x])/91287 - (33*Log[2*( 1 + x)])/49 - (67*Log[2*(2 + x)])/324 + (81*Log[8 + 6*x])/92
Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
\(\displaystyle \int \frac {1}{(5-2 x) (3 x+4) \left (x^2+3 x+2\right )^2} \, dx\) |
\(\Big \downarrow \) 1207 |
\(\displaystyle \int \left (-\frac {33}{49 (x+1)}-\frac {67}{324 (x+2)}+\frac {243}{92 (3 x+4)}+\frac {1}{7 (x+1)^2}-\frac {1}{18 (x+2)^2}+\frac {32}{91287 (5-2 x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{7 (x+1)}+\frac {1}{18 (x+2)}-\frac {16 \log (5-2 x)}{91287}-\frac {33}{49} \log (x+1)-\frac {67}{324} \log (x+2)+\frac {81}{92} \log (3 x+4)\) |
Input:
Int[1/((5 - 2*x)*(4 + 3*x)*(2 + 3*x + x^2)^2),x]
Output:
-1/7*1/(1 + x) + 1/(18*(2 + x)) - (16*Log[5 - 2*x])/91287 - (33*Log[1 + x] )/49 - (67*Log[2 + x])/324 + (81*Log[4 + 3*x])/92
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 = 1.81 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {16 \ln \left (-5+2 x \right )}{91287}-\frac {1}{7 \left (x +1\right )}-\frac {33 \ln \left (x +1\right )}{49}+\frac {81 \ln \left (3 x +4\right )}{92}+\frac {1}{36+18 x}-\frac {67 \ln \left (2+x \right )}{324}\) | \(44\) |
norman | \(\frac {-\frac {11 x}{126}-\frac {29}{126}}{x^{2}+3 x +2}-\frac {16 \ln \left (-5+2 x \right )}{91287}-\frac {33 \ln \left (x +1\right )}{49}-\frac {67 \ln \left (2+x \right )}{324}+\frac {81 \ln \left (3 x +4\right )}{92}\) | \(46\) |
risch | \(\frac {-\frac {11 x}{126}-\frac {29}{126}}{x^{2}+3 x +2}-\frac {16 \ln \left (-5+2 x \right )}{91287}-\frac {33 \ln \left (x +1\right )}{49}-\frac {67 \ln \left (2+x \right )}{324}+\frac {81 \ln \left (3 x +4\right )}{92}\) | \(46\) |
parallelrisch | \(-\frac {75509 \ln \left (2+x \right ) x^{2}+64 \ln \left (x -\frac {5}{2}\right ) x^{2}+245916 \ln \left (x +1\right ) x^{2}-321489 \ln \left (x +\frac {4}{3}\right ) x^{2}+84042+226527 \ln \left (2+x \right ) x +192 \ln \left (x -\frac {5}{2}\right ) x +737748 \ln \left (x +1\right ) x -964467 \ln \left (x +\frac {4}{3}\right ) x +151018 \ln \left (2+x \right )+128 \ln \left (x -\frac {5}{2}\right )+491832 \ln \left (x +1\right )-642978 \ln \left (x +\frac {4}{3}\right )+31878 x}{365148 \left (x^{2}+3 x +2\right )}\) | \(106\) |
Input:
int(1/(5-2*x)/(3*x+4)/(x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
-16/91287*ln(-5+2*x)-1/7/(x+1)-33/49*ln(x+1)+81/92*ln(3*x+4)+1/18/(2+x)-67 /324*ln(2+x)
Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(5-2 x) (4+3 x) \left (2+3 x+x^2\right )^2} \, dx=\frac {321489 \, {\left (x^{2} + 3 \, x + 2\right )} \log \left (3 \, x + 4\right ) - 64 \, {\left (x^{2} + 3 \, x + 2\right )} \log \left (2 \, x - 5\right ) - 75509 \, {\left (x^{2} + 3 \, x + 2\right )} \log \left (x + 2\right ) - 245916 \, {\left (x^{2} + 3 \, x + 2\right )} \log \left (x + 1\right ) - 31878 \, x - 84042}{365148 \, {\left (x^{2} + 3 \, x + 2\right )}} \] Input:
integrate(1/(5-2*x)/(4+3*x)/(x^2+3*x+2)^2,x, algorithm="fricas")
Output:
1/365148*(321489*(x^2 + 3*x + 2)*log(3*x + 4) - 64*(x^2 + 3*x + 2)*log(2*x - 5) - 75509*(x^2 + 3*x + 2)*log(x + 2) - 245916*(x^2 + 3*x + 2)*log(x + 1) - 31878*x - 84042)/(x^2 + 3*x + 2)
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(5-2 x) (4+3 x) \left (2+3 x+x^2\right )^2} \, dx=- \frac {11 x + 29}{126 x^{2} + 378 x + 252} - \frac {16 \log {\left (x - \frac {5}{2} \right )}}{91287} - \frac {33 \log {\left (x + 1 \right )}}{49} + \frac {81 \log {\left (x + \frac {4}{3} \right )}}{92} - \frac {67 \log {\left (x + 2 \right )}}{324} \] Input:
integrate(1/(5-2*x)/(4+3*x)/(x**2+3*x+2)**2,x)
Output:
-(11*x + 29)/(126*x**2 + 378*x + 252) - 16*log(x - 5/2)/91287 - 33*log(x + 1)/49 + 81*log(x + 4/3)/92 - 67*log(x + 2)/324
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(5-2 x) (4+3 x) \left (2+3 x+x^2\right )^2} \, dx=-\frac {11 \, x + 29}{126 \, {\left (x^{2} + 3 \, x + 2\right )}} + \frac {81}{92} \, \log \left (3 \, x + 4\right ) - \frac {16}{91287} \, \log \left (2 \, x - 5\right ) - \frac {67}{324} \, \log \left (x + 2\right ) - \frac {33}{49} \, \log \left (x + 1\right ) \] Input:
integrate(1/(5-2*x)/(4+3*x)/(x^2+3*x+2)^2,x, algorithm="maxima")
Output:
-1/126*(11*x + 29)/(x^2 + 3*x + 2) + 81/92*log(3*x + 4) - 16/91287*log(2*x - 5) - 67/324*log(x + 2) - 33/49*log(x + 1)
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(5-2 x) (4+3 x) \left (2+3 x+x^2\right )^2} \, dx=-\frac {11 \, x + 29}{126 \, {\left (x + 2\right )} {\left (x + 1\right )}} + \frac {81}{92} \, \log \left ({\left | 3 \, x + 4 \right |}\right ) - \frac {16}{91287} \, \log \left ({\left | 2 \, x - 5 \right |}\right ) - \frac {67}{324} \, \log \left ({\left | x + 2 \right |}\right ) - \frac {33}{49} \, \log \left ({\left | x + 1 \right |}\right ) \] Input:
integrate(1/(5-2*x)/(4+3*x)/(x^2+3*x+2)^2,x, algorithm="giac")
Output:
-1/126*(11*x + 29)/((x + 2)*(x + 1)) + 81/92*log(abs(3*x + 4)) - 16/91287* log(abs(2*x - 5)) - 67/324*log(abs(x + 2)) - 33/49*log(abs(x + 1))
Time = 13.81 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(5-2 x) (4+3 x) \left (2+3 x+x^2\right )^2} \, dx=\frac {81\,\ln \left (x+\frac {4}{3}\right )}{92}-\frac {67\,\ln \left (x+2\right )}{324}-\frac {16\,\ln \left (x-\frac {5}{2}\right )}{91287}-\frac {33\,\ln \left (x+1\right )}{49}-\frac {\frac {11\,x}{126}+\frac {29}{126}}{x^2+3\,x+2} \] Input:
int(-1/((2*x - 5)*(3*x + 4)*(3*x + x^2 + 2)^2),x)
Output:
(81*log(x + 4/3))/92 - (67*log(x + 2))/324 - (16*log(x - 5/2))/91287 - (33 *log(x + 1))/49 - ((11*x)/126 + 29/126)/(3*x + x^2 + 2)
Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.18 \[ \int \frac {1}{(5-2 x) (4+3 x) \left (2+3 x+x^2\right )^2} \, dx=\frac {321489 \,\mathrm {log}\left (3 x +4\right ) x^{2}+964467 \,\mathrm {log}\left (3 x +4\right ) x +642978 \,\mathrm {log}\left (3 x +4\right )-64 \,\mathrm {log}\left (2 x -5\right ) x^{2}-192 \,\mathrm {log}\left (2 x -5\right ) x -128 \,\mathrm {log}\left (2 x -5\right )-75509 \,\mathrm {log}\left (x +2\right ) x^{2}-226527 \,\mathrm {log}\left (x +2\right ) x -151018 \,\mathrm {log}\left (x +2\right )-245916 \,\mathrm {log}\left (x +1\right ) x^{2}-737748 \,\mathrm {log}\left (x +1\right ) x -491832 \,\mathrm {log}\left (x +1\right )+10626 x^{2}-62790}{365148 x^{2}+1095444 x +730296} \] Input:
int(1/(5-2*x)/(4+3*x)/(x^2+3*x+2)^2,x)
Output:
(321489*log(3*x + 4)*x**2 + 964467*log(3*x + 4)*x + 642978*log(3*x + 4) - 64*log(2*x - 5)*x**2 - 192*log(2*x - 5)*x - 128*log(2*x - 5) - 75509*log(x + 2)*x**2 - 226527*log(x + 2)*x - 151018*log(x + 2) - 245916*log(x + 1)*x **2 - 737748*log(x + 1)*x - 491832*log(x + 1) + 10626*x**2 - 62790)/(36514 8*(x**2 + 3*x + 2))