Integrand size = 25, antiderivative size = 63 \[ \int \frac {(5-2 x)^3}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=-\frac {343}{1+x}+\frac {729}{8 (2+x)}-\frac {12167}{8 (4+3 x)^2}-\frac {8993}{4 (4+3 x)}-4067 \log (1+x)-\frac {8505}{16} \log (2+x)+\frac {73577}{16} \log (4+3 x) \] Output:
-343/(1+x)+729/(16+8*x)-12167/8/(4+3*x)^2-8993/(16+12*x)-4067*ln(1+x)-8505 /16*ln(2+x)+73577/16*ln(4+3*x)
Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \frac {(5-2 x)^3}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=\frac {1}{16} \left (-\frac {24334}{(4+3 x)^2}-\frac {35972}{4+3 x}-\frac {9518+4030 x}{2+3 x+x^2}-65072 \log (-3 (1+x))-8505 \log (3 (2+x))+73577 \log (4+3 x)\right ) \] Input:
Integrate[(5 - 2*x)^3/((4 + 3*x)^3*(2 + 3*x + x^2)^2),x]
Output:
(-24334/(4 + 3*x)^2 - 35972/(4 + 3*x) - (9518 + 4030*x)/(2 + 3*x + x^2) - 65072*Log[-3*(1 + x)] - 8505*Log[3*(2 + x)] + 73577*Log[4 + 3*x])/16
Time = 0.36 (sec) , antiderivative size = 63, 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.
\(\displaystyle \int \frac {(5-2 x)^3}{(3 x+4)^3 \left (x^2+3 x+2\right )^2} \, dx\) |
\(\Big \downarrow \) 1207 |
\(\displaystyle \int \left (-\frac {8505}{16 (x+2)}+\frac {220731}{16 (3 x+4)}-\frac {729}{8 (x+2)^2}+\frac {26979}{4 (3 x+4)^2}+\frac {36501}{4 (3 x+4)^3}-\frac {4067}{x+1}+\frac {343}{(x+1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {343}{x+1}+\frac {729}{8 (x+2)}-\frac {8993}{4 (3 x+4)}-\frac {12167}{8 (3 x+4)^2}-4067 \log (x+1)-\frac {8505}{16} \log (x+2)+\frac {73577}{16} \log (3 x+4)\) |
Input:
Int[(5 - 2*x)^3/((4 + 3*x)^3*(2 + 3*x + x^2)^2),x]
Output:
-343/(1 + x) + 729/(8*(2 + x)) - 12167/(8*(4 + 3*x)^2) - 8993/(4*(4 + 3*x) ) - 4067*Log[1 + x] - (8505*Log[2 + x])/16 + (73577*Log[4 + 3*x])/16
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 = 1.85 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {343}{x +1}-4067 \ln \left (x +1\right )-\frac {12167}{8 \left (3 x +4\right )^{2}}-\frac {8993}{4 \left (3 x +4\right )}+\frac {73577 \ln \left (3 x +4\right )}{16}+\frac {729}{8 \left (2+x \right )}-\frac {8505 \ln \left (2+x \right )}{16}\) | \(54\) |
norman | \(\frac {-42147 x^{2}-\frac {506705}{8} x -\frac {72093}{8} x^{3}-\frac {122183}{4}}{\left (3 x +4\right )^{2} \left (x^{2}+3 x +2\right )}-4067 \ln \left (x +1\right )-\frac {8505 \ln \left (2+x \right )}{16}+\frac {73577 \ln \left (3 x +4\right )}{16}\) | \(55\) |
risch | \(\frac {-42147 x^{2}-\frac {506705}{8} x -\frac {72093}{8} x^{3}-\frac {122183}{4}}{\left (3 x +4\right )^{2} \left (x^{2}+3 x +2\right )}-4067 \ln \left (x +1\right )-\frac {8505 \ln \left (2+x \right )}{16}+\frac {73577 \ln \left (3 x +4\right )}{16}\) | \(56\) |
parallelrisch | \(-\frac {688905 \ln \left (2+x \right ) x^{4}+5270832 \ln \left (x +1\right ) x^{4}-5959737 \ln \left (x +\frac {4}{3}\right ) x^{4}+4398588+3903795 \ln \left (2+x \right ) x^{3}+29868048 \ln \left (x +1\right ) x^{3}-33771843 \ln \left (x +\frac {4}{3}\right ) x^{3}+8113770 \ln \left (2+x \right ) x^{2}+62078688 \ln \left (x +1\right ) x^{2}-70192458 \ln \left (x +\frac {4}{3}\right ) x^{2}+1297674 x^{3}+7348320 \ln \left (2+x \right ) x +56222208 \ln \left (x +1\right ) x -63570528 \ln \left (x +\frac {4}{3}\right ) x +6069168 x^{2}+2449440 \ln \left (2+x \right )+18740736 \ln \left (x +1\right )-21190176 \ln \left (x +\frac {4}{3}\right )+9120690 x}{144 \left (3 x +4\right )^{2} \left (x^{2}+3 x +2\right )}\) | \(155\) |
Input:
int((5-2*x)^3/(3*x+4)^3/(x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
-343/(x+1)-4067*ln(x+1)-12167/8/(3*x+4)^2-8993/4/(3*x+4)+73577/16*ln(3*x+4 )+729/8/(2+x)-8505/16*ln(2+x)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (53) = 106\).
Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.89 \[ \int \frac {(5-2 x)^3}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=-\frac {144186 \, x^{3} + 674352 \, x^{2} - 73577 \, {\left (9 \, x^{4} + 51 \, x^{3} + 106 \, x^{2} + 96 \, x + 32\right )} \log \left (3 \, x + 4\right ) + 8505 \, {\left (9 \, x^{4} + 51 \, x^{3} + 106 \, x^{2} + 96 \, x + 32\right )} \log \left (x + 2\right ) + 65072 \, {\left (9 \, x^{4} + 51 \, x^{3} + 106 \, x^{2} + 96 \, x + 32\right )} \log \left (x + 1\right ) + 1013410 \, x + 488732}{16 \, {\left (9 \, x^{4} + 51 \, x^{3} + 106 \, x^{2} + 96 \, x + 32\right )}} \] Input:
integrate((5-2*x)^3/(4+3*x)^3/(x^2+3*x+2)^2,x, algorithm="fricas")
Output:
-1/16*(144186*x^3 + 674352*x^2 - 73577*(9*x^4 + 51*x^3 + 106*x^2 + 96*x + 32)*log(3*x + 4) + 8505*(9*x^4 + 51*x^3 + 106*x^2 + 96*x + 32)*log(x + 2) + 65072*(9*x^4 + 51*x^3 + 106*x^2 + 96*x + 32)*log(x + 1) + 1013410*x + 48 8732)/(9*x^4 + 51*x^3 + 106*x^2 + 96*x + 32)
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int \frac {(5-2 x)^3}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=- \frac {72093 x^{3} + 337176 x^{2} + 506705 x + 244366}{72 x^{4} + 408 x^{3} + 848 x^{2} + 768 x + 256} - 4067 \log {\left (x + 1 \right )} + \frac {73577 \log {\left (x + \frac {4}{3} \right )}}{16} - \frac {8505 \log {\left (x + 2 \right )}}{16} \] Input:
integrate((5-2*x)**3/(4+3*x)**3/(x**2+3*x+2)**2,x)
Output:
-(72093*x**3 + 337176*x**2 + 506705*x + 244366)/(72*x**4 + 408*x**3 + 848* x**2 + 768*x + 256) - 4067*log(x + 1) + 73577*log(x + 4/3)/16 - 8505*log(x + 2)/16
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int \frac {(5-2 x)^3}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=-\frac {72093 \, x^{3} + 337176 \, x^{2} + 506705 \, x + 244366}{8 \, {\left (9 \, x^{4} + 51 \, x^{3} + 106 \, x^{2} + 96 \, x + 32\right )}} + \frac {73577}{16} \, \log \left (3 \, x + 4\right ) - \frac {8505}{16} \, \log \left (x + 2\right ) - 4067 \, \log \left (x + 1\right ) \] Input:
integrate((5-2*x)^3/(4+3*x)^3/(x^2+3*x+2)^2,x, algorithm="maxima")
Output:
-1/8*(72093*x^3 + 337176*x^2 + 506705*x + 244366)/(9*x^4 + 51*x^3 + 106*x^ 2 + 96*x + 32) + 73577/16*log(3*x + 4) - 8505/16*log(x + 2) - 4067*log(x + 1)
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \frac {(5-2 x)^3}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=-\frac {72093 \, x^{3} + 337176 \, x^{2} + 506705 \, x + 244366}{8 \, {\left (3 \, x + 4\right )}^{2} {\left (x + 2\right )} {\left (x + 1\right )}} + \frac {73577}{16} \, \log \left ({\left | 3 \, x + 4 \right |}\right ) - \frac {8505}{16} \, \log \left ({\left | x + 2 \right |}\right ) - 4067 \, \log \left ({\left | x + 1 \right |}\right ) \] Input:
integrate((5-2*x)^3/(4+3*x)^3/(x^2+3*x+2)^2,x, algorithm="giac")
Output:
-1/8*(72093*x^3 + 337176*x^2 + 506705*x + 244366)/((3*x + 4)^2*(x + 2)*(x + 1)) + 73577/16*log(abs(3*x + 4)) - 8505/16*log(abs(x + 2)) - 4067*log(ab s(x + 1))
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {(5-2 x)^3}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=\frac {73577\,\ln \left (x+\frac {4}{3}\right )}{16}-\frac {8505\,\ln \left (x+2\right )}{16}-4067\,\ln \left (x+1\right )-\frac {\frac {24031\,x^3}{24}+4683\,x^2+\frac {506705\,x}{72}+\frac {122183}{36}}{x^4+\frac {17\,x^3}{3}+\frac {106\,x^2}{9}+\frac {32\,x}{3}+\frac {32}{9}} \] Input:
int(-(2*x - 5)^3/((3*x + 4)^3*(3*x + x^2 + 2)^2),x)
Output:
(73577*log(x + 4/3))/16 - (8505*log(x + 2))/16 - 4067*log(x + 1) - ((50670 5*x)/72 + 4683*x^2 + (24031*x^3)/24 + 122183/36)/((32*x)/3 + (106*x^2)/9 + (17*x^3)/3 + x^4 + 32/9)
Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.67 \[ \int \frac {(5-2 x)^3}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=\frac {11257281 \,\mathrm {log}\left (3 x +4\right ) x^{4}+63791259 \,\mathrm {log}\left (3 x +4\right ) x^{3}+132585754 \,\mathrm {log}\left (3 x +4\right ) x^{2}+120077664 \,\mathrm {log}\left (3 x +4\right ) x +40025888 \,\mathrm {log}\left (3 x +4\right )-1301265 \,\mathrm {log}\left (x +2\right ) x^{4}-7373835 \,\mathrm {log}\left (x +2\right ) x^{3}-15326010 \,\mathrm {log}\left (x +2\right ) x^{2}-13880160 \,\mathrm {log}\left (x +2\right ) x -4626720 \,\mathrm {log}\left (x +2\right )-9956016 \,\mathrm {log}\left (x +1\right ) x^{4}-56417424 \,\mathrm {log}\left (x +1\right ) x^{3}-117259744 \,\mathrm {log}\left (x +1\right ) x^{2}-106197504 \,\mathrm {log}\left (x +1\right ) x -35399168 \,\mathrm {log}\left (x +1\right )+432558 x^{4}-6369412 x^{2}-12614018 x -6770460}{2448 x^{4}+13872 x^{3}+28832 x^{2}+26112 x +8704} \] Input:
int((5-2*x)^3/(4+3*x)^3/(x^2+3*x+2)^2,x)
Output:
(11257281*log(3*x + 4)*x**4 + 63791259*log(3*x + 4)*x**3 + 132585754*log(3 *x + 4)*x**2 + 120077664*log(3*x + 4)*x + 40025888*log(3*x + 4) - 1301265* log(x + 2)*x**4 - 7373835*log(x + 2)*x**3 - 15326010*log(x + 2)*x**2 - 138 80160*log(x + 2)*x - 4626720*log(x + 2) - 9956016*log(x + 1)*x**4 - 564174 24*log(x + 1)*x**3 - 117259744*log(x + 1)*x**2 - 106197504*log(x + 1)*x - 35399168*log(x + 1) + 432558*x**4 - 6369412*x**2 - 12614018*x - 6770460)/( 272*(9*x**4 + 51*x**3 + 106*x**2 + 96*x + 32))