Integrand size = 22, antiderivative size = 68 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {49}{3 (2+3 x)^3}+\frac {707}{2 (2+3 x)^2}+\frac {6934}{2+3 x}-\frac {605}{2 (3+5 x)^2}+\frac {7480}{3+5 x}-57110 \log (2+3 x)+57110 \log (3+5 x) \] Output:
49/3/(2+3*x)^3+707/2/(2+3*x)^2+6934/(2+3*x)-605/2/(3+5*x)^2+7480/(3+5*x)-5 7110*ln(2+3*x)+57110*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.03 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {49}{3 (2+3 x)^3}+\frac {707}{2 (2+3 x)^2}+\frac {6934}{2+3 x}-\frac {605}{2 (3+5 x)^2}+\frac {7480}{3+5 x}-57110 \log (5 (2+3 x))+57110 \log (3+5 x) \] Input:
Integrate[(1 - 2*x)^2/((2 + 3*x)^4*(3 + 5*x)^3),x]
Output:
49/(3*(2 + 3*x)^3) + 707/(2*(2 + 3*x)^2) + 6934/(2 + 3*x) - 605/(2*(3 + 5* x)^2) + 7480/(3 + 5*x) - 57110*Log[5*(2 + 3*x)] + 57110*Log[3 + 5*x]
Time = 0.24 (sec) , antiderivative size = 68, 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.
\(\displaystyle \int \frac {(1-2 x)^2}{(3 x+2)^4 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {285550}{5 x+3}-\frac {37400}{(5 x+3)^2}+\frac {3025}{(5 x+3)^3}-\frac {171330}{3 x+2}-\frac {20802}{(3 x+2)^2}-\frac {2121}{(3 x+2)^3}-\frac {147}{(3 x+2)^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6934}{3 x+2}+\frac {7480}{5 x+3}+\frac {707}{2 (3 x+2)^2}-\frac {605}{2 (5 x+3)^2}+\frac {49}{3 (3 x+2)^3}-57110 \log (3 x+2)+57110 \log (5 x+3)\) |
Input:
Int[(1 - 2*x)^2/((2 + 3*x)^4*(3 + 5*x)^3),x]
Output:
49/(3*(2 + 3*x)^3) + 707/(2*(2 + 3*x)^2) + 6934/(2 + 3*x) - 605/(2*(3 + 5* x)^2) + 7480/(3 + 5*x) - 57110*Log[2 + 3*x] + 57110*Log[3 + 5*x]
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 = 53, normalized size of antiderivative = 0.78
method | result | size |
norman | \(\frac {2569950 x^{4}+6596205 x^{3}+\frac {5416693}{2} x +\frac {19029052}{3} x^{2}+433234}{\left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-57110 \ln \left (2+3 x \right )+57110 \ln \left (3+5 x \right )\) | \(53\) |
risch | \(\frac {2569950 x^{4}+6596205 x^{3}+\frac {5416693}{2} x +\frac {19029052}{3} x^{2}+433234}{\left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-57110 \ln \left (2+3 x \right )+57110 \ln \left (3+5 x \right )\) | \(54\) |
default | \(\frac {49}{3 \left (2+3 x \right )^{3}}+\frac {707}{2 \left (2+3 x \right )^{2}}+\frac {6934}{2+3 x}-\frac {605}{2 \left (3+5 x \right )^{2}}+\frac {7480}{3+5 x}-57110 \ln \left (2+3 x \right )+57110 \ln \left (3+5 x \right )\) | \(63\) |
parallelrisch | \(-\frac {2775546000 \ln \left (\frac {2}{3}+x \right ) x^{5}-2775546000 \ln \left (x +\frac {3}{5}\right ) x^{5}+8881747200 \ln \left (\frac {2}{3}+x \right ) x^{4}-8881747200 \ln \left (x +\frac {3}{5}\right ) x^{4}+292432950 x^{5}+11361234960 \ln \left (\frac {2}{3}+x \right ) x^{3}-11361234960 \ln \left (x +\frac {3}{5}\right ) x^{3}+750749040 x^{4}+7261650720 \ln \left (\frac {2}{3}+x \right ) x^{2}-7261650720 \ln \left (x +\frac {3}{5}\right ) x^{2}+722098782 x^{3}+2319122880 \ln \left (\frac {2}{3}+x \right ) x -2319122880 \ln \left (x +\frac {3}{5}\right ) x +308393996 x^{2}+296058240 \ln \left (\frac {2}{3}+x \right )-296058240 \ln \left (x +\frac {3}{5}\right )+49343028 x}{72 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}\) | \(139\) |
Input:
int((1-2*x)^2/(2+3*x)^4/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
(2569950*x^4+6596205*x^3+5416693/2*x+19029052/3*x^2+433234)/(2+3*x)^3/(3+5 *x)^2-57110*ln(2+3*x)+57110*ln(3+5*x)
Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.69 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {15419700 \, x^{4} + 39577230 \, x^{3} + 38058104 \, x^{2} + 342660 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 342660 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (3 \, x + 2\right ) + 16250079 \, x + 2599404}{6 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \] Input:
integrate((1-2*x)^2/(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")
Output:
1/6*(15419700*x^4 + 39577230*x^3 + 38058104*x^2 + 342660*(675*x^5 + 2160*x ^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(5*x + 3) - 342660*(675*x^5 + 21 60*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(3*x + 2) + 16250079*x + 259 9404)/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {15419700 x^{4} + 39577230 x^{3} + 38058104 x^{2} + 16250079 x + 2599404}{4050 x^{5} + 12960 x^{4} + 16578 x^{3} + 10596 x^{2} + 3384 x + 432} + 57110 \log {\left (x + \frac {3}{5} \right )} - 57110 \log {\left (x + \frac {2}{3} \right )} \] Input:
integrate((1-2*x)**2/(2+3*x)**4/(3+5*x)**3,x)
Output:
(15419700*x**4 + 39577230*x**3 + 38058104*x**2 + 16250079*x + 2599404)/(40 50*x**5 + 12960*x**4 + 16578*x**3 + 10596*x**2 + 3384*x + 432) + 57110*log (x + 3/5) - 57110*log(x + 2/3)
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {15419700 \, x^{4} + 39577230 \, x^{3} + 38058104 \, x^{2} + 16250079 \, x + 2599404}{6 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} + 57110 \, \log \left (5 \, x + 3\right ) - 57110 \, \log \left (3 \, x + 2\right ) \] Input:
integrate((1-2*x)^2/(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")
Output:
1/6*(15419700*x^4 + 39577230*x^3 + 38058104*x^2 + 16250079*x + 2599404)/(6 75*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72) + 57110*log(5*x + 3) - 57110*log(3*x + 2)
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {15419700 \, x^{4} + 39577230 \, x^{3} + 38058104 \, x^{2} + 16250079 \, x + 2599404}{6 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{3}} + 57110 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 57110 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \] Input:
integrate((1-2*x)^2/(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")
Output:
1/6*(15419700*x^4 + 39577230*x^3 + 38058104*x^2 + 16250079*x + 2599404)/(( 5*x + 3)^2*(3*x + 2)^3) + 57110*log(abs(5*x + 3)) - 57110*log(abs(3*x + 2) )
Time = 1.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {\frac {11422\,x^4}{3}+\frac {439747\,x^3}{45}+\frac {19029052\,x^2}{2025}+\frac {5416693\,x}{1350}+\frac {433234}{675}}{x^5+\frac {16\,x^4}{5}+\frac {307\,x^3}{75}+\frac {1766\,x^2}{675}+\frac {188\,x}{225}+\frac {8}{75}}-114220\,\mathrm {atanh}\left (30\,x+19\right ) \] Input:
int((2*x - 1)^2/((3*x + 2)^4*(5*x + 3)^3),x)
Output:
((5416693*x)/1350 + (19029052*x^2)/2025 + (439747*x^3)/45 + (11422*x^4)/3 + 433234/675)/((188*x)/225 + (1766*x^2)/675 + (307*x^3)/75 + (16*x^4)/5 + x^5 + 8/75) - 114220*atanh(30*x + 19)
Time = 0.16 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.50 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {308394000 \,\mathrm {log}\left (5 x +3\right ) x^{5}+986860800 \,\mathrm {log}\left (5 x +3\right ) x^{4}+1262359440 \,\mathrm {log}\left (5 x +3\right ) x^{3}+806850080 \,\mathrm {log}\left (5 x +3\right ) x^{2}+257680320 \,\mathrm {log}\left (5 x +3\right ) x +32895360 \,\mathrm {log}\left (5 x +3\right )-308394000 \,\mathrm {log}\left (3 x +2\right ) x^{5}-986860800 \,\mathrm {log}\left (3 x +2\right ) x^{4}-1262359440 \,\mathrm {log}\left (3 x +2\right ) x^{3}-806850080 \,\mathrm {log}\left (3 x +2\right ) x^{2}-257680320 \,\mathrm {log}\left (3 x +2\right ) x -32895360 \,\mathrm {log}\left (3 x +2\right )-6424875 x^{5}+26470485 x^{3}+33934762 x^{2}+16298432 x +2780552}{5400 x^{5}+17280 x^{4}+22104 x^{3}+14128 x^{2}+4512 x +576} \] Input:
int((1-2*x)^2/(2+3*x)^4/(3+5*x)^3,x)
Output:
(308394000*log(5*x + 3)*x**5 + 986860800*log(5*x + 3)*x**4 + 1262359440*lo g(5*x + 3)*x**3 + 806850080*log(5*x + 3)*x**2 + 257680320*log(5*x + 3)*x + 32895360*log(5*x + 3) - 308394000*log(3*x + 2)*x**5 - 986860800*log(3*x + 2)*x**4 - 1262359440*log(3*x + 2)*x**3 - 806850080*log(3*x + 2)*x**2 - 25 7680320*log(3*x + 2)*x - 32895360*log(3*x + 2) - 6424875*x**5 + 26470485*x **3 + 33934762*x**2 + 16298432*x + 2780552)/(8*(675*x**5 + 2160*x**4 + 276 3*x**3 + 1766*x**2 + 564*x + 72))