Integrand size = 22, antiderivative size = 65 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^3} \, dx=\frac {121}{686 (1-2 x)^2}+\frac {319}{2401 (1-2 x)}-\frac {1}{686 (2+3 x)^2}+\frac {64}{2401 (2+3 x)}-\frac {829 \log (1-2 x)}{16807}+\frac {829 \log (2+3 x)}{16807} \] Output:
121/686/(1-2*x)^2+319/(2401-4802*x)-1/686/(2+3*x)^2+64/(4802+7203*x)-829/1 6807*ln(1-2*x)+829/16807*ln(2+3*x)
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^3} \, dx=\frac {-\frac {7 \left (-6189-12104 x+2487 x^2+9948 x^3\right )}{\left (-2+x+6 x^2\right )^2}-1658 \log (1-2 x)+1658 \log (2+3 x)}{33614} \] Input:
Integrate[(3 + 5*x)^2/((1 - 2*x)^3*(2 + 3*x)^3),x]
Output:
((-7*(-6189 - 12104*x + 2487*x^2 + 9948*x^3))/(-2 + x + 6*x^2)^2 - 1658*Lo g[1 - 2*x] + 1658*Log[2 + 3*x])/33614
Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {99, 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.
\(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x)^3 (3 x+2)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {2487}{16807 (3 x+2)}-\frac {192}{2401 (3 x+2)^2}+\frac {3}{343 (3 x+2)^3}-\frac {1658}{16807 (2 x-1)}+\frac {638}{2401 (2 x-1)^2}-\frac {242}{343 (2 x-1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {319}{2401 (1-2 x)}+\frac {64}{2401 (3 x+2)}+\frac {121}{686 (1-2 x)^2}-\frac {1}{686 (3 x+2)^2}-\frac {829 \log (1-2 x)}{16807}+\frac {829 \log (3 x+2)}{16807}\) |
Input:
Int[(3 + 5*x)^2/((1 - 2*x)^3*(2 + 3*x)^3),x]
Output:
121/(686*(1 - 2*x)^2) + 319/(2401*(1 - 2*x)) - 1/(686*(2 + 3*x)^2) + 64/(2 401*(2 + 3*x)) - (829*Log[1 - 2*x])/16807 + (829*Log[2 + 3*x])/16807
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74
method | result | size |
norman | \(\frac {-\frac {4974}{2401} x^{3}-\frac {2487}{4802} x^{2}+\frac {6052}{2401} x +\frac {6189}{4802}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{2}}-\frac {829 \ln \left (-1+2 x \right )}{16807}+\frac {829 \ln \left (2+3 x \right )}{16807}\) | \(48\) |
risch | \(\frac {-\frac {4974}{2401} x^{3}-\frac {2487}{4802} x^{2}+\frac {6052}{2401} x +\frac {6189}{4802}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{2}}-\frac {829 \ln \left (-1+2 x \right )}{16807}+\frac {829 \ln \left (2+3 x \right )}{16807}\) | \(49\) |
default | \(-\frac {1}{686 \left (2+3 x \right )^{2}}+\frac {64}{2401 \left (2+3 x \right )}+\frac {829 \ln \left (2+3 x \right )}{16807}+\frac {121}{686 \left (-1+2 x \right )^{2}}-\frac {319}{2401 \left (-1+2 x \right )}-\frac {829 \ln \left (-1+2 x \right )}{16807}\) | \(54\) |
parallelrisch | \(\frac {119376 \ln \left (\frac {2}{3}+x \right ) x^{4}-119376 \ln \left (x -\frac {1}{2}\right ) x^{4}+256102+39792 \ln \left (\frac {2}{3}+x \right ) x^{3}-39792 \ln \left (x -\frac {1}{2}\right ) x^{3}+1525104 x^{4}-76268 \ln \left (\frac {2}{3}+x \right ) x^{2}+76268 \ln \left (x -\frac {1}{2}\right ) x^{2}+369096 x^{3}-13264 \ln \left (\frac {2}{3}+x \right ) x +13264 \ln \left (x -\frac {1}{2}\right ) x -1009190 x^{2}+13264 \ln \left (\frac {2}{3}+x \right )-13264 \ln \left (x -\frac {1}{2}\right )}{67228 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{2}}\) | \(114\) |
Input:
int((3+5*x)^2/(1-2*x)^3/(2+3*x)^3,x,method=_RETURNVERBOSE)
Output:
(-4974/2401*x^3-2487/4802*x^2+6052/2401*x+6189/4802)/(-1+2*x)^2/(2+3*x)^2- 829/16807*ln(-1+2*x)+829/16807*ln(2+3*x)
Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.46 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^3} \, dx=-\frac {69636 \, x^{3} + 17409 \, x^{2} - 1658 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 1658 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (2 \, x - 1\right ) - 84728 \, x - 43323}{33614 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \] Input:
integrate((3+5*x)^2/(1-2*x)^3/(2+3*x)^3,x, algorithm="fricas")
Output:
-1/33614*(69636*x^3 + 17409*x^2 - 1658*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4 )*log(3*x + 2) + 1658*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(2*x - 1) - 84728*x - 43323)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^3} \, dx=- \frac {9948 x^{3} + 2487 x^{2} - 12104 x - 6189}{172872 x^{4} + 57624 x^{3} - 110446 x^{2} - 19208 x + 19208} - \frac {829 \log {\left (x - \frac {1}{2} \right )}}{16807} + \frac {829 \log {\left (x + \frac {2}{3} \right )}}{16807} \] Input:
integrate((3+5*x)**2/(1-2*x)**3/(2+3*x)**3,x)
Output:
-(9948*x**3 + 2487*x**2 - 12104*x - 6189)/(172872*x**4 + 57624*x**3 - 1104 46*x**2 - 19208*x + 19208) - 829*log(x - 1/2)/16807 + 829*log(x + 2/3)/168 07
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^3} \, dx=-\frac {9948 \, x^{3} + 2487 \, x^{2} - 12104 \, x - 6189}{4802 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} + \frac {829}{16807} \, \log \left (3 \, x + 2\right ) - \frac {829}{16807} \, \log \left (2 \, x - 1\right ) \] Input:
integrate((3+5*x)^2/(1-2*x)^3/(2+3*x)^3,x, algorithm="maxima")
Output:
-1/4802*(9948*x^3 + 2487*x^2 - 12104*x - 6189)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4) + 829/16807*log(3*x + 2) - 829/16807*log(2*x - 1)
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^3} \, dx=-\frac {9948 \, x^{3} + 2487 \, x^{2} - 12104 \, x - 6189}{4802 \, {\left (6 \, x^{2} + x - 2\right )}^{2}} + \frac {829}{16807} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {829}{16807} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \] Input:
integrate((3+5*x)^2/(1-2*x)^3/(2+3*x)^3,x, algorithm="giac")
Output:
-1/4802*(9948*x^3 + 2487*x^2 - 12104*x - 6189)/(6*x^2 + x - 2)^2 + 829/168 07*log(abs(3*x + 2)) - 829/16807*log(abs(2*x - 1))
Time = 1.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^3} \, dx=\frac {1658\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{16807}+\frac {-\frac {829\,x^3}{14406}-\frac {829\,x^2}{57624}+\frac {1513\,x}{21609}+\frac {2063}{57624}}{x^4+\frac {x^3}{3}-\frac {23\,x^2}{36}-\frac {x}{9}+\frac {1}{9}} \] Input:
int(-(5*x + 3)^2/((2*x - 1)^3*(3*x + 2)^3),x)
Output:
(1658*atanh((12*x)/7 + 1/7))/16807 + ((1513*x)/21609 - (829*x^2)/57624 - ( 829*x^3)/14406 + 2063/57624)/(x^3/3 - (23*x^2)/36 - x/9 + x^4 + 1/9)
Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.12 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^3} \, dx=\frac {59688 \,\mathrm {log}\left (3 x +2\right ) x^{4}+19896 \,\mathrm {log}\left (3 x +2\right ) x^{3}-38134 \,\mathrm {log}\left (3 x +2\right ) x^{2}-6632 \,\mathrm {log}\left (3 x +2\right ) x +6632 \,\mathrm {log}\left (3 x +2\right )-59688 \,\mathrm {log}\left (2 x -1\right ) x^{4}-19896 \,\mathrm {log}\left (2 x -1\right ) x^{3}+38134 \,\mathrm {log}\left (2 x -1\right ) x^{2}+6632 \,\mathrm {log}\left (2 x -1\right ) x -6632 \,\mathrm {log}\left (2 x -1\right )+208908 x^{4}-150878 x^{2}+61516 x +66535}{1210104 x^{4}+403368 x^{3}-773122 x^{2}-134456 x +134456} \] Input:
int((3+5*x)^2/(1-2*x)^3/(2+3*x)^3,x)
Output:
(59688*log(3*x + 2)*x**4 + 19896*log(3*x + 2)*x**3 - 38134*log(3*x + 2)*x* *2 - 6632*log(3*x + 2)*x + 6632*log(3*x + 2) - 59688*log(2*x - 1)*x**4 - 1 9896*log(2*x - 1)*x**3 + 38134*log(2*x - 1)*x**2 + 6632*log(2*x - 1)*x - 6 632*log(2*x - 1) + 208908*x**4 - 150878*x**2 + 61516*x + 66535)/(33614*(36 *x**4 + 12*x**3 - 23*x**2 - 4*x + 4))