Integrand size = 25, antiderivative size = 65 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {64 x}{27}-\frac {32 x^2}{27}+\frac {3}{(1+x)^2}+\frac {125}{1+x}-\frac {265625}{486 (2+3 x)^2}+\frac {175000}{243 (2+3 x)}-1311 \log (1+x)+\frac {109375}{81} \log (2+3 x) \] Output:
64/27*x-32/27*x^2+3/(1+x)^2+125/(1+x)-265625/486/(2+3*x)^2+175000/(486+729 *x)-1311*ln(1+x)+109375/81*ln(2+3*x)
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.32 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {695303+2973334 x+3955871 x^2+1718574 x^3+40464 x^4-6912 x^5-5184 x^6+656250 \left (2+5 x+3 x^2\right )^2 \log (-4-6 x)-637146 \left (2+5 x+3 x^2\right )^2 \log (-2 (1+x))}{486 \left (2+5 x+3 x^2\right )^2} \] Input:
Integrate[((5 - x)*(3 + 2*x)^6)/(2 + 5*x + 3*x^2)^3,x]
Output:
(695303 + 2973334*x + 3955871*x^2 + 1718574*x^3 + 40464*x^4 - 6912*x^5 - 5 184*x^6 + 656250*(2 + 5*x + 3*x^2)^2*Log[-4 - 6*x] - 637146*(2 + 5*x + 3*x ^2)^2*Log[-2*(1 + x)])/(486*(2 + 5*x + 3*x^2)^2)
Time = 0.38 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.12, 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.
\(\displaystyle \int \frac {(5-x) (2 x+3)^6}{\left (3 x^2+5 x+2\right )^3} \, dx\) |
\(\Big \downarrow \) 1207 |
\(\displaystyle 27 \int \left (-\frac {64 x}{729}-\frac {437}{9 (x+1)}+\frac {109375}{729 (3 x+2)}-\frac {125}{27 (x+1)^2}-\frac {175000}{2187 (3 x+2)^2}-\frac {2}{9 (x+1)^3}+\frac {265625}{2187 (3 x+2)^3}+\frac {64}{729}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 27 \left (-\frac {32 x^2}{729}+\frac {64 x}{729}+\frac {125}{27 (x+1)}+\frac {175000}{6561 (3 x+2)}+\frac {1}{9 (x+1)^2}-\frac {265625}{13122 (3 x+2)^2}-\frac {437}{9} \log (x+1)+\frac {109375 \log (3 x+2)}{2187}\right )\) |
Input:
Int[((5 - x)*(3 + 2*x)^6)/(2 + 5*x + 3*x^2)^3,x]
Output:
27*((64*x)/729 - (32*x^2)/729 + 1/(9*(1 + x)^2) + 125/(27*(1 + x)) - 26562 5/(13122*(2 + 3*x)^2) + 175000/(6561*(2 + 3*x)) - (437*Log[1 + x])/9 + (10 9375*Log[2 + 3*x])/2187)
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 = 1.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {32 x^{2}}{27}+\frac {64 x}{27}+\frac {\frac {266125}{81} x^{3}+\frac {3823247}{486} x^{2}+\frac {1454123}{243} x +\frac {683207}{486}}{\left (3 x^{2}+5 x +2\right )^{2}}-1311 \ln \left (x +1\right )+\frac {109375 \ln \left (3 x +2\right )}{81}\) | \(53\) |
norman | \(\frac {\frac {87983}{27} x^{3}+\frac {480569}{81} x +\frac {1263173}{162} x^{2}-\frac {128}{9} x^{5}-\frac {32}{3} x^{6}+\frac {225773}{162}}{\left (3 x^{2}+5 x +2\right )^{2}}-1311 \ln \left (x +1\right )+\frac {109375 \ln \left (3 x +2\right )}{81}\) | \(54\) |
default | \(-\frac {32 x^{2}}{27}+\frac {64 x}{27}-\frac {265625}{486 \left (3 x +2\right )^{2}}+\frac {175000}{243 \left (3 x +2\right )}+\frac {109375 \ln \left (3 x +2\right )}{81}+\frac {3}{\left (x +1\right )^{2}}+\frac {125}{x +1}-1311 \ln \left (x +1\right )\) | \(56\) |
parallelrisch | \(-\frac {6912 x^{6}+7645752 \ln \left (x +1\right ) x^{4}-7875000 \ln \left (x +\frac {2}{3}\right ) x^{4}+9216 x^{5}+25485840 \ln \left (x +1\right ) x^{3}-26250000 \ln \left (x +\frac {2}{3}\right ) x^{3}+2031957 x^{4}+31432536 \ln \left (x +1\right ) x^{2}-32375000 \ln \left (x +\frac {2}{3}\right ) x^{2}+4661598 x^{3}+16990560 \ln \left (x +1\right ) x -17500000 \ln \left (x +\frac {2}{3}\right ) x +3300909 x^{2}+3398112 \ln \left (x +1\right )-3500000 \ln \left (x +\frac {2}{3}\right )+670908 x}{648 \left (3 x^{2}+5 x +2\right )^{2}}\) | \(124\) |
Input:
int((5-x)*(2*x+3)^6/(3*x^2+5*x+2)^3,x,method=_RETURNVERBOSE)
Output:
-32/27*x^2+64/27*x+9*(266125/729*x^3+3823247/4374*x^2+1454123/2187*x+68320 7/4374)/(3*x^2+5*x+2)^2-1311*ln(x+1)+109375/81*ln(3*x+2)
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.66 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx=-\frac {5184 \, x^{6} + 6912 \, x^{5} - 13248 \, x^{4} - 1627854 \, x^{3} - 3843983 \, x^{2} - 656250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 637146 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) - 2912854 \, x - 683207}{486 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \] Input:
integrate((5-x)*(3+2*x)^6/(3*x^2+5*x+2)^3,x, algorithm="fricas")
Output:
-1/486*(5184*x^6 + 6912*x^5 - 13248*x^4 - 1627854*x^3 - 3843983*x^2 - 6562 50*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(3*x + 2) + 637146*(9*x^4 + 30* x^3 + 37*x^2 + 20*x + 4)*log(x + 1) - 2912854*x - 683207)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx=- \frac {32 x^{2}}{27} + \frac {64 x}{27} - \frac {- 1596750 x^{3} - 3823247 x^{2} - 2908246 x - 683207}{4374 x^{4} + 14580 x^{3} + 17982 x^{2} + 9720 x + 1944} + \frac {109375 \log {\left (x + \frac {2}{3} \right )}}{81} - 1311 \log {\left (x + 1 \right )} \] Input:
integrate((5-x)*(3+2*x)**6/(3*x**2+5*x+2)**3,x)
Output:
-32*x**2/27 + 64*x/27 - (-1596750*x**3 - 3823247*x**2 - 2908246*x - 683207 )/(4374*x**4 + 14580*x**3 + 17982*x**2 + 9720*x + 1944) + 109375*log(x + 2 /3)/81 - 1311*log(x + 1)
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx=-\frac {32}{27} \, x^{2} + \frac {64}{27} \, x + \frac {1596750 \, x^{3} + 3823247 \, x^{2} + 2908246 \, x + 683207}{486 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + \frac {109375}{81} \, \log \left (3 \, x + 2\right ) - 1311 \, \log \left (x + 1\right ) \] Input:
integrate((5-x)*(3+2*x)^6/(3*x^2+5*x+2)^3,x, algorithm="maxima")
Output:
-32/27*x^2 + 64/27*x + 1/486*(1596750*x^3 + 3823247*x^2 + 2908246*x + 6832 07)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4) + 109375/81*log(3*x + 2) - 1311*l og(x + 1)
Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx=-\frac {32}{27} \, x^{2} + \frac {64}{27} \, x + \frac {1596750 \, x^{3} + 3823247 \, x^{2} + 2908246 \, x + 683207}{486 \, {\left (3 \, x + 2\right )}^{2} {\left (x + 1\right )}^{2}} + \frac {109375}{81} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 1311 \, \log \left ({\left | x + 1 \right |}\right ) \] Input:
integrate((5-x)*(3+2*x)^6/(3*x^2+5*x+2)^3,x, algorithm="giac")
Output:
-32/27*x^2 + 64/27*x + 1/486*(1596750*x^3 + 3823247*x^2 + 2908246*x + 6832 07)/((3*x + 2)^2*(x + 1)^2) + 109375/81*log(abs(3*x + 2)) - 1311*log(abs(x + 1))
Time = 11.99 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {64\,x}{27}-1311\,\ln \left (x+1\right )+\frac {109375\,\ln \left (x+\frac {2}{3}\right )}{81}-\frac {32\,x^2}{27}+\frac {\frac {266125\,x^3}{729}+\frac {3823247\,x^2}{4374}+\frac {1454123\,x}{2187}+\frac {683207}{4374}}{x^4+\frac {10\,x^3}{3}+\frac {37\,x^2}{9}+\frac {20\,x}{9}+\frac {4}{9}} \] Input:
int(-((2*x + 3)^6*(x - 5))/(5*x + 3*x^2 + 2)^3,x)
Output:
(64*x)/27 - 1311*log(x + 1) + (109375*log(x + 2/3))/81 - (32*x^2)/27 + ((1 454123*x)/2187 + (3823247*x^2)/4374 + (266125*x^3)/729 + 683207/4374)/((20 *x)/9 + (37*x^2)/9 + (10*x^3)/3 + x^4 + 4/9)
Time = 0.24 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.12 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {9843750 \,\mathrm {log}\left (3 x +2\right ) x^{4}+32812500 \,\mathrm {log}\left (3 x +2\right ) x^{3}+40468750 \,\mathrm {log}\left (3 x +2\right ) x^{2}+21875000 \,\mathrm {log}\left (3 x +2\right ) x +4375000 \,\mathrm {log}\left (3 x +2\right )-9557190 \,\mathrm {log}\left (x +1\right ) x^{4}-31857300 \,\mathrm {log}\left (x +1\right ) x^{3}-39290670 \,\mathrm {log}\left (x +1\right ) x^{2}-21238200 \,\mathrm {log}\left (x +1\right ) x -4247640 \,\mathrm {log}\left (x +1\right )-8640 x^{6}-11520 x^{5}-791847 x^{4}+3060494 x^{2}+3046030 x +776933}{7290 x^{4}+24300 x^{3}+29970 x^{2}+16200 x +3240} \] Input:
int((5-x)*(3+2*x)^6/(3*x^2+5*x+2)^3,x)
Output:
(9843750*log(3*x + 2)*x**4 + 32812500*log(3*x + 2)*x**3 + 40468750*log(3*x + 2)*x**2 + 21875000*log(3*x + 2)*x + 4375000*log(3*x + 2) - 9557190*log( x + 1)*x**4 - 31857300*log(x + 1)*x**3 - 39290670*log(x + 1)*x**2 - 212382 00*log(x + 1)*x - 4247640*log(x + 1) - 8640*x**6 - 11520*x**5 - 791847*x** 4 + 3060494*x**2 + 3046030*x + 776933)/(810*(9*x**4 + 30*x**3 + 37*x**2 + 20*x + 4))