Integrand size = 25, antiderivative size = 65 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^2 (4+3 x)^4} \, dx=\frac {3969}{559682 (5-2 x)}-\frac {4}{42849 (4+3 x)^3}+\frac {38}{328509 (4+3 x)^2}+\frac {1907}{7555707 (4+3 x)}-\frac {630 \log (5-2 x)}{6436343}+\frac {630 \log (4+3 x)}{6436343} \] Output:
3969/(2798410-1119364*x)-4/42849/(4+3*x)^3+38/328509/(4+3*x)^2+1907/(30222 828+22667121*x)-630/6436343*ln(5-2*x)+630/6436343*ln(4+3*x)
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^2 (4+3 x)^4} \, dx=\frac {23 \left (\frac {321489}{5-2 x}-\frac {4232}{(4+3 x)^3}+\frac {5244}{(4+3 x)^2}+\frac {11442}{4+3 x}\right )-102060 \log (5-2 x)+102060 \log (8+6 x)}{1042687566} \] Input:
Integrate[(2 + 3*x + x^2)^2/((5 - 2*x)^2*(4 + 3*x)^4),x]
Output:
(23*(321489/(5 - 2*x) - 4232/(4 + 3*x)^3 + 5244/(4 + 3*x)^2 + 11442/(4 + 3 *x)) - 102060*Log[5 - 2*x] + 102060*Log[8 + 6*x])/1042687566
Time = 0.34 (sec) , antiderivative size = 65, 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 = {1195, 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.
\(\displaystyle \int \frac {\left (x^2+3 x+2\right )^2}{(5-2 x)^2 (3 x+4)^4} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {1890}{6436343 (3 x+4)}-\frac {1907}{2518569 (3 x+4)^2}-\frac {76}{109503 (3 x+4)^3}+\frac {4}{4761 (3 x+4)^4}-\frac {1260}{6436343 (2 x-5)}+\frac {3969}{279841 (2 x-5)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3969}{559682 (5-2 x)}+\frac {1907}{7555707 (3 x+4)}+\frac {38}{328509 (3 x+4)^2}-\frac {4}{42849 (3 x+4)^3}-\frac {630 \log (5-2 x)}{6436343}+\frac {630 \log (3 x+4)}{6436343}\) |
Input:
Int[(2 + 3*x + x^2)^2/((5 - 2*x)^2*(4 + 3*x)^4),x]
Output:
3969/(559682*(5 - 2*x)) - 4/(42849*(4 + 3*x)^3) + 38/(328509*(4 + 3*x)^2) + 1907/(7555707*(4 + 3*x)) - (630*Log[5 - 2*x])/6436343 + (630*Log[4 + 3*x ])/6436343
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 1.62 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74
method | result | size |
norman | \(\frac {-\frac {23673242}{22667121} x -\frac {1925279}{2518569} x^{2}-\frac {313861}{1679046} x^{3}-\frac {10787188}{22667121}}{\left (-5+2 x \right ) \left (3 x +4\right )^{3}}-\frac {630 \ln \left (-5+2 x \right )}{6436343}+\frac {630 \ln \left (3 x +4\right )}{6436343}\) | \(48\) |
risch | \(\frac {-\frac {23673242}{22667121} x -\frac {1925279}{2518569} x^{2}-\frac {313861}{1679046} x^{3}-\frac {10787188}{22667121}}{\left (-5+2 x \right ) \left (3 x +4\right )^{3}}-\frac {630 \ln \left (-5+2 x \right )}{6436343}+\frac {630 \ln \left (3 x +4\right )}{6436343}\) | \(49\) |
default | \(-\frac {3969}{559682 \left (-5+2 x \right )}-\frac {630 \ln \left (-5+2 x \right )}{6436343}-\frac {4}{42849 \left (3 x +4\right )^{3}}+\frac {38}{328509 \left (3 x +4\right )^{2}}+\frac {1907}{7555707 \left (3 x +4\right )}+\frac {630 \ln \left (3 x +4\right )}{6436343}\) | \(54\) |
parallelrisch | \(-\frac {5511240 \ln \left (x -\frac {5}{2}\right ) x^{4}-5511240 \ln \left (x +\frac {4}{3}\right ) x^{4}+496210648+8266860 \ln \left (x -\frac {5}{2}\right ) x^{3}-8266860 \ln \left (x +\frac {4}{3}\right ) x^{3}-25719120 \ln \left (x -\frac {5}{2}\right ) x^{2}+25719120 \ln \left (x +\frac {4}{3}\right ) x^{2}+194907681 x^{3}-60419520 \ln \left (x -\frac {5}{2}\right ) x +60419520 \ln \left (x +\frac {4}{3}\right ) x +797065506 x^{2}-32659200 \ln \left (x -\frac {5}{2}\right )+32659200 \ln \left (x +\frac {4}{3}\right )+1088969132 x}{1042687566 \left (-5+2 x \right ) \left (3 x +4\right )^{3}}\) | \(112\) |
Input:
int((x^2+3*x+2)^2/(5-2*x)^2/(3*x+4)^4,x,method=_RETURNVERBOSE)
Output:
(-23673242/22667121*x-1925279/2518569*x^2-313861/1679046*x^3-10787188/2266 7121)/(-5+2*x)/(3*x+4)^3-630/6436343*ln(-5+2*x)+630/6436343*ln(3*x+4)
Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.46 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^2 (4+3 x)^4} \, dx=-\frac {194907681 \, x^{3} + 797065506 \, x^{2} - 102060 \, {\left (54 \, x^{4} + 81 \, x^{3} - 252 \, x^{2} - 592 \, x - 320\right )} \log \left (3 \, x + 4\right ) + 102060 \, {\left (54 \, x^{4} + 81 \, x^{3} - 252 \, x^{2} - 592 \, x - 320\right )} \log \left (2 \, x - 5\right ) + 1088969132 \, x + 496210648}{1042687566 \, {\left (54 \, x^{4} + 81 \, x^{3} - 252 \, x^{2} - 592 \, x - 320\right )}} \] Input:
integrate((x^2+3*x+2)^2/(5-2*x)^2/(4+3*x)^4,x, algorithm="fricas")
Output:
-1/1042687566*(194907681*x^3 + 797065506*x^2 - 102060*(54*x^4 + 81*x^3 - 2 52*x^2 - 592*x - 320)*log(3*x + 4) + 102060*(54*x^4 + 81*x^3 - 252*x^2 - 5 92*x - 320)*log(2*x - 5) + 1088969132*x + 496210648)/(54*x^4 + 81*x^3 - 25 2*x^2 - 592*x - 320)
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^2 (4+3 x)^4} \, dx=\frac {- 8474247 x^{3} - 34655022 x^{2} - 47346484 x - 21574376}{2448049068 x^{4} + 3672073602 x^{3} - 11424228984 x^{2} - 26837871264 x - 14506957440} - \frac {630 \log {\left (x - \frac {5}{2} \right )}}{6436343} + \frac {630 \log {\left (x + \frac {4}{3} \right )}}{6436343} \] Input:
integrate((x**2+3*x+2)**2/(5-2*x)**2/(4+3*x)**4,x)
Output:
(-8474247*x**3 - 34655022*x**2 - 47346484*x - 21574376)/(2448049068*x**4 + 3672073602*x**3 - 11424228984*x**2 - 26837871264*x - 14506957440) - 630*l og(x - 5/2)/6436343 + 630*log(x + 4/3)/6436343
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^2 (4+3 x)^4} \, dx=-\frac {8474247 \, x^{3} + 34655022 \, x^{2} + 47346484 \, x + 21574376}{45334242 \, {\left (54 \, x^{4} + 81 \, x^{3} - 252 \, x^{2} - 592 \, x - 320\right )}} + \frac {630}{6436343} \, \log \left (3 \, x + 4\right ) - \frac {630}{6436343} \, \log \left (2 \, x - 5\right ) \] Input:
integrate((x^2+3*x+2)^2/(5-2*x)^2/(4+3*x)^4,x, algorithm="maxima")
Output:
-1/45334242*(8474247*x^3 + 34655022*x^2 + 47346484*x + 21574376)/(54*x^4 + 81*x^3 - 252*x^2 - 592*x - 320) + 630/6436343*log(3*x + 4) - 630/6436343* log(2*x - 5)
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^2 (4+3 x)^4} \, dx=-\frac {3969}{559682 \, {\left (2 \, x - 5\right )}} - \frac {2 \, {\left (\frac {92598}{2 \, x - 5} + \frac {360249}{{\left (2 \, x - 5\right )}^{2}} + 5933\right )}}{19309029 \, {\left (\frac {23}{2 \, x - 5} + 3\right )}^{3}} + \frac {630}{6436343} \, \log \left ({\left | -\frac {23}{2 \, x - 5} - 3 \right |}\right ) \] Input:
integrate((x^2+3*x+2)^2/(5-2*x)^2/(4+3*x)^4,x, algorithm="giac")
Output:
-3969/559682/(2*x - 5) - 2/19309029*(92598/(2*x - 5) + 360249/(2*x - 5)^2 + 5933)/(23/(2*x - 5) + 3)^3 + 630/6436343*log(abs(-23/(2*x - 5) - 3))
Time = 12.68 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^2 (4+3 x)^4} \, dx=\frac {1260\,\mathrm {atanh}\left (\frac {12\,x}{23}-\frac {7}{23}\right )}{6436343}+\frac {\frac {313861\,x^3}{90668484}+\frac {1925279\,x^2}{136002726}+\frac {11836621\,x}{612012267}+\frac {5393594}{612012267}}{-x^4-\frac {3\,x^3}{2}+\frac {14\,x^2}{3}+\frac {296\,x}{27}+\frac {160}{27}} \] Input:
int((3*x + x^2 + 2)^2/((2*x - 5)^2*(3*x + 4)^4),x)
Output:
(1260*atanh((12*x)/23 - 7/23))/6436343 + ((11836621*x)/612012267 + (192527 9*x^2)/136002726 + (313861*x^3)/90668484 + 5393594/612012267)/((296*x)/27 + (14*x^2)/3 - (3*x^3)/2 - x^4 + 160/27)
Time = 0.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.12 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^2 (4+3 x)^4} \, dx=\frac {8266860 \,\mathrm {log}\left (3 x +4\right ) x^{4}+12400290 \,\mathrm {log}\left (3 x +4\right ) x^{3}-38578680 \,\mathrm {log}\left (3 x +4\right ) x^{2}-90629280 \,\mathrm {log}\left (3 x +4\right ) x -48988800 \,\mathrm {log}\left (3 x +4\right )-8266860 \,\mathrm {log}\left (2 x -5\right ) x^{4}-12400290 \,\mathrm {log}\left (2 x -5\right ) x^{3}+38578680 \,\mathrm {log}\left (2 x -5\right ) x^{2}+90629280 \,\mathrm {log}\left (2 x -5\right ) x +48988800 \,\mathrm {log}\left (2 x -5\right )+194907681 x^{4}-2105167437 x^{2}-3770219386 x -1899324452}{84457692846 x^{4}+126686539269 x^{3}-394135899948 x^{2}-925906558608 x -500490031680} \] Input:
int((x^2+3*x+2)^2/(5-2*x)^2/(4+3*x)^4,x)
Output:
(8266860*log(3*x + 4)*x**4 + 12400290*log(3*x + 4)*x**3 - 38578680*log(3*x + 4)*x**2 - 90629280*log(3*x + 4)*x - 48988800*log(3*x + 4) - 8266860*log (2*x - 5)*x**4 - 12400290*log(2*x - 5)*x**3 + 38578680*log(2*x - 5)*x**2 + 90629280*log(2*x - 5)*x + 48988800*log(2*x - 5) + 194907681*x**4 - 210516 7437*x**2 - 3770219386*x - 1899324452)/(1564031349*(54*x**4 + 81*x**3 - 25 2*x**2 - 592*x - 320))