Integrand size = 25, antiderivative size = 52 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \left (2+3 x+x^2\right )^2} \, dx=-\frac {343}{1+x}-\frac {729}{4 (2+x)}-\frac {12167}{4 (4+3 x)}-3038 \log (1+x)+\frac {3159}{4} \log (2+x)+\frac {8993}{4} \log (4+3 x) \] Output:
-343/(1+x)-729/(8+4*x)-12167/(16+12*x)-3038*ln(1+x)+3159/4*ln(2+x)+8993/4* ln(4+3*x)
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \left (2+3 x+x^2\right )^2} \, dx=\frac {1}{4} \left (-\frac {12167}{4+3 x}-\frac {3473+2101 x}{2+3 x+x^2}-12152 \log (-3 (1+x))+3159 \log (3 (2+x))+8993 \log (4+3 x)\right ) \] Input:
Integrate[(5 - 2*x)^3/((4 + 3*x)^2*(2 + 3*x + x^2)^2),x]
Output:
(-12167/(4 + 3*x) - (3473 + 2101*x)/(2 + 3*x + x^2) - 12152*Log[-3*(1 + x) ] + 3159*Log[3*(2 + x)] + 8993*Log[4 + 3*x])/4
Time = 0.35 (sec) , antiderivative size = 52, 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.
\(\displaystyle \int \frac {(5-2 x)^3}{(3 x+4)^2 \left (x^2+3 x+2\right )^2} \, dx\) |
\(\Big \downarrow \) 1207 |
\(\displaystyle \int \left (\frac {3159}{4 (x+2)}+\frac {26979}{4 (3 x+4)}+\frac {729}{4 (x+2)^2}+\frac {36501}{4 (3 x+4)^2}-\frac {3038}{x+1}+\frac {343}{(x+1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {343}{x+1}-\frac {729}{4 (x+2)}-\frac {12167}{4 (3 x+4)}-3038 \log (x+1)+\frac {3159}{4} \log (x+2)+\frac {8993}{4} \log (3 x+4)\) |
Input:
Int[(5 - 2*x)^3/((4 + 3*x)^2*(2 + 3*x + x^2)^2),x]
Output:
-343/(1 + x) - 729/(4*(2 + x)) - 12167/(4*(4 + 3*x)) - 3038*Log[1 + x] + ( 3159*Log[2 + x])/4 + (8993*Log[4 + 3*x])/4
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 = 45, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {343}{x +1}-3038 \ln \left (x +1\right )-\frac {12167}{4 \left (3 x +4\right )}+\frac {8993 \ln \left (3 x +4\right )}{4}-\frac {729}{4 \left (2+x \right )}+\frac {3159 \ln \left (2+x \right )}{4}\) | \(45\) |
norman | \(\frac {-13831 x -\frac {9235}{2} x^{2}-\frac {19113}{2}}{\left (3 x +4\right ) \left (x^{2}+3 x +2\right )}-3038 \ln \left (x +1\right )+\frac {3159 \ln \left (2+x \right )}{4}+\frac {8993 \ln \left (3 x +4\right )}{4}\) | \(50\) |
risch | \(\frac {-13831 x -\frac {9235}{2} x^{2}-\frac {19113}{2}}{\left (3 x +4\right ) \left (x^{2}+3 x +2\right )}-3038 \ln \left (x +1\right )+\frac {3159 \ln \left (2+x \right )}{4}+\frac {8993 \ln \left (3 x +4\right )}{4}\) | \(51\) |
parallelrisch | \(\frac {28431 \ln \left (2+x \right ) x^{3}-109368 \ln \left (x +1\right ) x^{3}+80937 \ln \left (x +\frac {4}{3}\right ) x^{3}-114678+123201 \ln \left (2+x \right ) x^{2}-473928 \ln \left (x +1\right ) x^{2}+350727 \ln \left (x +\frac {4}{3}\right ) x^{2}+170586 \ln \left (2+x \right ) x -656208 \ln \left (x +1\right ) x +485622 \ln \left (x +\frac {4}{3}\right ) x -55410 x^{2}+75816 \ln \left (2+x \right )-291648 \ln \left (x +1\right )+215832 \ln \left (x +\frac {4}{3}\right )-165972 x}{12 \left (3 x +4\right ) \left (x^{2}+3 x +2\right )}\) | \(123\) |
Input:
int((5-2*x)^3/(3*x+4)^2/(x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
-343/(x+1)-3038*ln(x+1)-12167/4/(3*x+4)+8993/4*ln(3*x+4)-729/4/(2+x)+3159/ 4*ln(2+x)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (44) = 88\).
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.81 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \left (2+3 x+x^2\right )^2} \, dx=-\frac {18470 \, x^{2} - 8993 \, {\left (3 \, x^{3} + 13 \, x^{2} + 18 \, x + 8\right )} \log \left (3 \, x + 4\right ) - 3159 \, {\left (3 \, x^{3} + 13 \, x^{2} + 18 \, x + 8\right )} \log \left (x + 2\right ) + 12152 \, {\left (3 \, x^{3} + 13 \, x^{2} + 18 \, x + 8\right )} \log \left (x + 1\right ) + 55324 \, x + 38226}{4 \, {\left (3 \, x^{3} + 13 \, x^{2} + 18 \, x + 8\right )}} \] Input:
integrate((5-2*x)^3/(4+3*x)^2/(x^2+3*x+2)^2,x, algorithm="fricas")
Output:
-1/4*(18470*x^2 - 8993*(3*x^3 + 13*x^2 + 18*x + 8)*log(3*x + 4) - 3159*(3* x^3 + 13*x^2 + 18*x + 8)*log(x + 2) + 12152*(3*x^3 + 13*x^2 + 18*x + 8)*lo g(x + 1) + 55324*x + 38226)/(3*x^3 + 13*x^2 + 18*x + 8)
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.94 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \left (2+3 x+x^2\right )^2} \, dx=- \frac {9235 x^{2} + 27662 x + 19113}{6 x^{3} + 26 x^{2} + 36 x + 16} - 3038 \log {\left (x + 1 \right )} + \frac {8993 \log {\left (x + \frac {4}{3} \right )}}{4} + \frac {3159 \log {\left (x + 2 \right )}}{4} \] Input:
integrate((5-2*x)**3/(4+3*x)**2/(x**2+3*x+2)**2,x)
Output:
-(9235*x**2 + 27662*x + 19113)/(6*x**3 + 26*x**2 + 36*x + 16) - 3038*log(x + 1) + 8993*log(x + 4/3)/4 + 3159*log(x + 2)/4
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \left (2+3 x+x^2\right )^2} \, dx=-\frac {9235 \, x^{2} + 27662 \, x + 19113}{2 \, {\left (3 \, x^{3} + 13 \, x^{2} + 18 \, x + 8\right )}} + \frac {8993}{4} \, \log \left (3 \, x + 4\right ) + \frac {3159}{4} \, \log \left (x + 2\right ) - 3038 \, \log \left (x + 1\right ) \] Input:
integrate((5-2*x)^3/(4+3*x)^2/(x^2+3*x+2)^2,x, algorithm="maxima")
Output:
-1/2*(9235*x^2 + 27662*x + 19113)/(3*x^3 + 13*x^2 + 18*x + 8) + 8993/4*log (3*x + 4) + 3159/4*log(x + 2) - 3038*log(x + 1)
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.48 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \left (2+3 x+x^2\right )^2} \, dx=-\frac {12167}{4 \, {\left (3 \, x + 4\right )}} + \frac {3 \, {\left (\frac {6217}{3 \, x + 4} + 2015\right )}}{8 \, {\left (\frac {2}{3 \, x + 4} + 1\right )} {\left (\frac {1}{3 \, x + 4} - 1\right )}} - 3038 \, \log \left ({\left | -\frac {1}{3 \, x + 4} + 1 \right |}\right ) + \frac {3159}{4} \, \log \left ({\left | -\frac {2}{3 \, x + 4} - 1 \right |}\right ) \] Input:
integrate((5-2*x)^3/(4+3*x)^2/(x^2+3*x+2)^2,x, algorithm="giac")
Output:
-12167/4/(3*x + 4) + 3/8*(6217/(3*x + 4) + 2015)/((2/(3*x + 4) + 1)*(1/(3* x + 4) - 1)) - 3038*log(abs(-1/(3*x + 4) + 1)) + 3159/4*log(abs(-2/(3*x + 4) - 1))
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \left (2+3 x+x^2\right )^2} \, dx=\frac {3159\,\ln \left (x+2\right )}{4}-3038\,\ln \left (x+1\right )+\frac {8993\,\ln \left (x+\frac {4}{3}\right )}{4}-\frac {\frac {9235\,x^2}{6}+\frac {13831\,x}{3}+\frac {6371}{2}}{x^3+\frac {13\,x^2}{3}+6\,x+\frac {8}{3}} \] Input:
int(-(2*x - 5)^3/((3*x + 4)^2*(3*x + x^2 + 2)^2),x)
Output:
(3159*log(x + 2))/4 - 3038*log(x + 1) + (8993*log(x + 4/3))/4 - ((13831*x) /3 + (9235*x^2)/6 + 6371/2)/(6*x + (13*x^2)/3 + x^3 + 8/3)
Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.48 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \left (2+3 x+x^2\right )^2} \, dx=\frac {350727 \,\mathrm {log}\left (3 x +4\right ) x^{3}+1519817 \,\mathrm {log}\left (3 x +4\right ) x^{2}+2104362 \,\mathrm {log}\left (3 x +4\right ) x +935272 \,\mathrm {log}\left (3 x +4\right )+123201 \,\mathrm {log}\left (x +2\right ) x^{3}+533871 \,\mathrm {log}\left (x +2\right ) x^{2}+739206 \,\mathrm {log}\left (x +2\right ) x +328536 \,\mathrm {log}\left (x +2\right )-473928 \,\mathrm {log}\left (x +1\right ) x^{3}-2053688 \,\mathrm {log}\left (x +1\right ) x^{2}-2843568 \,\mathrm {log}\left (x +1\right ) x -1263808 \,\mathrm {log}\left (x +1\right )+55410 x^{3}-386752 x -349178}{156 x^{3}+676 x^{2}+936 x +416} \] Input:
int((5-2*x)^3/(4+3*x)^2/(x^2+3*x+2)^2,x)
Output:
(350727*log(3*x + 4)*x**3 + 1519817*log(3*x + 4)*x**2 + 2104362*log(3*x + 4)*x + 935272*log(3*x + 4) + 123201*log(x + 2)*x**3 + 533871*log(x + 2)*x* *2 + 739206*log(x + 2)*x + 328536*log(x + 2) - 473928*log(x + 1)*x**3 - 20 53688*log(x + 1)*x**2 - 2843568*log(x + 1)*x - 1263808*log(x + 1) + 55410* x**3 - 386752*x - 349178)/(52*(3*x**3 + 13*x**2 + 18*x + 8))