Integrand size = 22, antiderivative size = 101 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {49}{6 (2+3 x)^6}+\frac {707}{5 (2+3 x)^5}+\frac {3467}{2 (2+3 x)^4}+\frac {57110}{3 (2+3 x)^3}+\frac {424975}{2 (2+3 x)^2}+\frac {2958125}{2+3 x}-\frac {75625}{2 (3+5 x)^2}+\frac {1615625}{3+5 x}-19637500 \log (2+3 x)+19637500 \log (3+5 x) \]
49/6/(2+3*x)^6+707/5/(2+3*x)^5+3467/2/(2+3*x)^4+57110/3/(2+3*x)^3+424975/2 /(2+3*x)^2+2958125/(2+3*x)-75625/2/(3+5*x)^2+1615625/(3+5*x)-19637500*ln(2 +3*x)+19637500*ln(3+5*x)
Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {49}{6 (2+3 x)^6}+\frac {707}{5 (2+3 x)^5}+\frac {3467}{2 (2+3 x)^4}+\frac {57110}{3 (2+3 x)^3}+\frac {424975}{2 (2+3 x)^2}+\frac {2958125}{2+3 x}-\frac {75625}{2 (3+5 x)^2}+\frac {1615625}{3+5 x}-19637500 \log (5 (2+3 x))+19637500 \log (3+5 x) \]
49/(6*(2 + 3*x)^6) + 707/(5*(2 + 3*x)^5) + 3467/(2*(2 + 3*x)^4) + 57110/(3 *(2 + 3*x)^3) + 424975/(2*(2 + 3*x)^2) + 2958125/(2 + 3*x) - 75625/(2*(3 + 5*x)^2) + 1615625/(3 + 5*x) - 19637500*Log[5*(2 + 3*x)] + 19637500*Log[3 + 5*x]
Time = 0.24 (sec) , antiderivative size = 101, 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.
| \(\displaystyle \int \frac {(1-2 x)^2}{(3 x+2)^7 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {98187500}{5 x+3}-\frac {8078125}{(5 x+3)^2}+\frac {378125}{(5 x+3)^3}-\frac {58912500}{3 x+2}-\frac {8874375}{(3 x+2)^2}-\frac {1274925}{(3 x+2)^3}-\frac {171330}{(3 x+2)^4}-\frac {20802}{(3 x+2)^5}-\frac {2121}{(3 x+2)^6}-\frac {147}{(3 x+2)^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2958125}{3 x+2}+\frac {1615625}{5 x+3}+\frac {424975}{2 (3 x+2)^2}-\frac {75625}{2 (5 x+3)^2}+\frac {57110}{3 (3 x+2)^3}+\frac {3467}{2 (3 x+2)^4}+\frac {707}{5 (3 x+2)^5}+\frac {49}{6 (3 x+2)^6}-19637500 \log (3 x+2)+19637500 \log (5 x+3)\) |
49/(6*(2 + 3*x)^6) + 707/(5*(2 + 3*x)^5) + 3467/(2*(2 + 3*x)^4) + 57110/(3 *(2 + 3*x)^3) + 424975/(2*(2 + 3*x)^2) + 2958125/(2 + 3*x) - 75625/(2*(3 + 5*x)^2) + 1615625/(3 + 5*x) - 19637500*Log[2 + 3*x] + 19637500*Log[3 + 5* x]
3.14.36.3.1 Defintions of rubi rules used
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 = 2.34 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67
| method | result | size |
| norman | \(\frac {23859562500 x^{7}+59018836277 x^{2}+108958668750 x^{6}+150974686710 x^{3}+213180772500 x^{5}+231644539125 x^{4}+\frac {64065488324}{5} x +\frac {11917538647}{10}}{\left (2+3 x \right )^{6} \left (3+5 x \right )^{2}}-19637500 \ln \left (2+3 x \right )+19637500 \ln \left (3+5 x \right )\) | \(68\) |
| risch | \(\frac {23859562500 x^{7}+59018836277 x^{2}+108958668750 x^{6}+150974686710 x^{3}+213180772500 x^{5}+231644539125 x^{4}+\frac {64065488324}{5} x +\frac {11917538647}{10}}{\left (2+3 x \right )^{6} \left (3+5 x \right )^{2}}-19637500 \ln \left (2+3 x \right )+19637500 \ln \left (3+5 x \right )\) | \(69\) |
| default | \(\frac {49}{6 \left (2+3 x \right )^{6}}+\frac {707}{5 \left (2+3 x \right )^{5}}+\frac {3467}{2 \left (2+3 x \right )^{4}}+\frac {57110}{3 \left (2+3 x \right )^{3}}+\frac {424975}{2 \left (2+3 x \right )^{2}}+\frac {2958125}{2+3 x}-\frac {75625}{2 \left (3+5 x \right )^{2}}+\frac {1615625}{3+5 x}-19637500 \ln \left (2+3 x \right )+19637500 \ln \left (3+5 x \right )\) | \(90\) |
| parallelrisch | \(-\frac {10858751999040 x -4334451840000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+13356264960000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-803547648000000 \ln \left (x +\frac {3}{5}\right ) x +4334451840000000 \ln \left (\frac {2}{3}+x \right ) x^{2}+803547648000000 \ln \left (\frac {2}{3}+x \right ) x +2109513943260468 x^{5}+1941116265513027 x^{6}+991994057576190 x^{7}+537608767988160 x^{3}+1375060690648980 x^{4}+116731583997520 x^{2}+217197141841575 x^{8}+25714882080000000 \ln \left (\frac {2}{3}+x \right ) x^{4}+65152512000000 \ln \left (\frac {2}{3}+x \right )+10719624240000000 \ln \left (\frac {2}{3}+x \right ) x^{7}-10719624240000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-65152512000000 \ln \left (x +\frac {3}{5}\right )+31676336928000000 \ln \left (\frac {2}{3}+x \right ) x^{5}-13356264960000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-31676336928000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-25714882080000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+24380273592000000 \ln \left (\frac {2}{3}+x \right ) x^{6}-24380273592000000 \ln \left (x +\frac {3}{5}\right ) x^{6}+2061466200000000 \ln \left (\frac {2}{3}+x \right ) x^{8}-2061466200000000 \ln \left (x +\frac {3}{5}\right ) x^{8}}{5760 \left (2+3 x \right )^{6} \left (3+5 x \right )^{2}}\) | \(208\) |
(23859562500*x^7+59018836277*x^2+108958668750*x^6+150974686710*x^3+2131807 72500*x^5+231644539125*x^4+64065488324/5*x+11917538647/10)/(2+3*x)^6/(3+5* x)^2-19637500*ln(2+3*x)+19637500*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.73 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {238595625000 \, x^{7} + 1089586687500 \, x^{6} + 2131807725000 \, x^{5} + 2316445391250 \, x^{4} + 1509746867100 \, x^{3} + 590188362770 \, x^{2} + 196375000 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )} \log \left (5 \, x + 3\right ) - 196375000 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )} \log \left (3 \, x + 2\right ) + 128130976648 \, x + 11917538647}{10 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )}} \]
1/10*(238595625000*x^7 + 1089586687500*x^6 + 2131807725000*x^5 + 231644539 1250*x^4 + 1509746867100*x^3 + 590188362770*x^2 + 196375000*(18225*x^8 + 9 4770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576)*log(5*x + 3) - 196375000*(18225*x^8 + 94770*x^7 + 215541*x^ 6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576)*log(3 *x + 2) + 128130976648*x + 11917538647)/(18225*x^8 + 94770*x^7 + 215541*x^ 6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576)
Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {238595625000 x^{7} + 1089586687500 x^{6} + 2131807725000 x^{5} + 2316445391250 x^{4} + 1509746867100 x^{3} + 590188362770 x^{2} + 128130976648 x + 11917538647}{182250 x^{8} + 947700 x^{7} + 2155410 x^{6} + 2800440 x^{5} + 2273400 x^{4} + 1180800 x^{3} + 383200 x^{2} + 71040 x + 5760} + 19637500 \log {\left (x + \frac {3}{5} \right )} - 19637500 \log {\left (x + \frac {2}{3} \right )} \]
(238595625000*x**7 + 1089586687500*x**6 + 2131807725000*x**5 + 23164453912 50*x**4 + 1509746867100*x**3 + 590188362770*x**2 + 128130976648*x + 119175 38647)/(182250*x**8 + 947700*x**7 + 2155410*x**6 + 2800440*x**5 + 2273400* x**4 + 1180800*x**3 + 383200*x**2 + 71040*x + 5760) + 19637500*log(x + 3/5 ) - 19637500*log(x + 2/3)
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {238595625000 \, x^{7} + 1089586687500 \, x^{6} + 2131807725000 \, x^{5} + 2316445391250 \, x^{4} + 1509746867100 \, x^{3} + 590188362770 \, x^{2} + 128130976648 \, x + 11917538647}{10 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )}} + 19637500 \, \log \left (5 \, x + 3\right ) - 19637500 \, \log \left (3 \, x + 2\right ) \]
1/10*(238595625000*x^7 + 1089586687500*x^6 + 2131807725000*x^5 + 231644539 1250*x^4 + 1509746867100*x^3 + 590188362770*x^2 + 128130976648*x + 1191753 8647)/(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 1180 80*x^3 + 38320*x^2 + 7104*x + 576) + 19637500*log(5*x + 3) - 19637500*log( 3*x + 2)
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {238595625000 \, x^{7} + 1089586687500 \, x^{6} + 2131807725000 \, x^{5} + 2316445391250 \, x^{4} + 1509746867100 \, x^{3} + 590188362770 \, x^{2} + 128130976648 \, x + 11917538647}{10 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{6}} + 19637500 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 19637500 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]
1/10*(238595625000*x^7 + 1089586687500*x^6 + 2131807725000*x^5 + 231644539 1250*x^4 + 1509746867100*x^3 + 590188362770*x^2 + 128130976648*x + 1191753 8647)/((5*x + 3)^2*(3*x + 2)^6) + 19637500*log(abs(5*x + 3)) - 19637500*lo g(abs(3*x + 2))
Time = 1.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^3} \, dx=\frac {\frac {3927500\,x^7}{3}+\frac {53806750\,x^6}{9}+\frac {947470100\,x^5}{81}+\frac {114392365\,x^4}{9}+\frac {3354993038\,x^3}{405}+\frac {59018836277\,x^2}{18225}+\frac {64065488324\,x}{91125}+\frac {11917538647}{182250}}{x^8+\frac {26\,x^7}{5}+\frac {887\,x^6}{75}+\frac {10372\,x^5}{675}+\frac {1684\,x^4}{135}+\frac {2624\,x^3}{405}+\frac {7664\,x^2}{3645}+\frac {2368\,x}{6075}+\frac {64}{2025}}-39275000\,\mathrm {atanh}\left (30\,x+19\right ) \]
((64065488324*x)/91125 + (59018836277*x^2)/18225 + (3354993038*x^3)/405 + (114392365*x^4)/9 + (947470100*x^5)/81 + (53806750*x^6)/9 + (3927500*x^7)/ 3 + 11917538647/182250)/((2368*x)/6075 + (7664*x^2)/3645 + (2624*x^3)/405 + (1684*x^4)/135 + (10372*x^5)/675 + (887*x^6)/75 + (26*x^7)/5 + x^8 + 64/ 2025) - 39275000*atanh(30*x + 19)