Integrand size = 22, antiderivative size = 75 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {343}{12 (2+3 x)^4}+\frac {539}{(2+3 x)^3}+\frac {7854}{(2+3 x)^2}+\frac {128634}{2+3 x}-\frac {6655}{2 (3+5 x)^2}+\frac {103455}{3+5 x}-953535 \log (2+3 x)+953535 \log (3+5 x) \]
343/12/(2+3*x)^4+539/(2+3*x)^3+7854/(2+3*x)^2+128634/(2+3*x)-6655/2/(3+5*x )^2+103455/(3+5*x)-953535*ln(2+3*x)+953535*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {343}{12 (2+3 x)^4}+\frac {539}{(2+3 x)^3}+\frac {7854}{(2+3 x)^2}+\frac {128634}{2+3 x}-\frac {6655}{2 (3+5 x)^2}+\frac {103455}{3+5 x}-953535 \log (5 (2+3 x))+953535 \log (3+5 x) \]
343/(12*(2 + 3*x)^4) + 539/(2 + 3*x)^3 + 7854/(2 + 3*x)^2 + 128634/(2 + 3* x) - 6655/(2*(3 + 5*x)^2) + 103455/(3 + 5*x) - 953535*Log[5*(2 + 3*x)] + 9 53535*Log[3 + 5*x]
Time = 0.21 (sec) , antiderivative size = 75, 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.
\(\displaystyle \int \frac {(1-2 x)^3}{(3 x+2)^5 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {4767675}{5 x+3}-\frac {517275}{(5 x+3)^2}+\frac {33275}{(5 x+3)^3}-\frac {2860605}{3 x+2}-\frac {385902}{(3 x+2)^2}-\frac {47124}{(3 x+2)^3}-\frac {4851}{(3 x+2)^4}-\frac {343}{(3 x+2)^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {128634}{3 x+2}+\frac {103455}{5 x+3}+\frac {7854}{(3 x+2)^2}-\frac {6655}{2 (5 x+3)^2}+\frac {539}{(3 x+2)^3}+\frac {343}{12 (3 x+2)^4}-953535 \log (3 x+2)+953535 \log (5 x+3)\) |
343/(12*(2 + 3*x)^4) + 539/(2 + 3*x)^3 + 7854/(2 + 3*x)^2 + 128634/(2 + 3* x) - 6655/(2*(3 + 5*x)^2) + 103455/(3 + 5*x) - 953535*Log[2 + 3*x] + 95353 5*Log[3 + 5*x]
3.15.30.3.1 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 = 2.49 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77
method | result | size |
norman | \(\frac {128727225 x^{5}+537984447 x^{3}+\frac {224280077}{2} x +\frac {832436055}{2} x^{4}+\frac {4169655991}{12} x^{2}+\frac {57867805}{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-953535 \ln \left (2+3 x \right )+953535 \ln \left (3+5 x \right )\) | \(58\) |
risch | \(\frac {128727225 x^{5}+537984447 x^{3}+\frac {224280077}{2} x +\frac {832436055}{2} x^{4}+\frac {4169655991}{12} x^{2}+\frac {57867805}{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-953535 \ln \left (2+3 x \right )+953535 \ln \left (3+5 x \right )\) | \(59\) |
default | \(\frac {343}{12 \left (2+3 x \right )^{4}}+\frac {539}{\left (2+3 x \right )^{3}}+\frac {7854}{\left (2+3 x \right )^{2}}+\frac {128634}{2+3 x}-\frac {6655}{2 \left (3+5 x \right )^{2}}+\frac {103455}{3+5 x}-953535 \ln \left (2+3 x \right )+953535 \ln \left (3+5 x \right )\) | \(72\) |
parallelrisch | \(-\frac {13181667744 x -2869209699840 \ln \left (x +\frac {3}{5}\right ) x^{2}+5944932195840 \ln \left (\frac {2}{3}+x \right ) x^{3}-738173399040 \ln \left (x +\frac {3}{5}\right ) x +2869209699840 \ln \left (\frac {2}{3}+x \right ) x^{2}+738173399040 \ln \left (\frac {2}{3}+x \right ) x +378958031550 x^{5}+117182305125 x^{6}+316482079848 x^{3}+489913569405 x^{4}+102157925752 x^{2}+6925318741440 \ln \left (\frac {2}{3}+x \right ) x^{4}+79090007040 \ln \left (\frac {2}{3}+x \right )-79090007040 \ln \left (x +\frac {3}{5}\right )+4300519132800 \ln \left (\frac {2}{3}+x \right ) x^{5}-5944932195840 \ln \left (x +\frac {3}{5}\right ) x^{3}-4300519132800 \ln \left (x +\frac {3}{5}\right ) x^{5}-6925318741440 \ln \left (x +\frac {3}{5}\right ) x^{4}+1112203224000 \ln \left (\frac {2}{3}+x \right ) x^{6}-1112203224000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{576 \left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}\) | \(162\) |
(128727225*x^5+537984447*x^3+224280077/2*x+832436055/2*x^4+4169655991/12*x ^2+57867805/4)/(2+3*x)^4/(3+5*x)^2-953535*ln(2+3*x)+953535*ln(3+5*x)
Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.80 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {1544726700 \, x^{5} + 4994616330 \, x^{4} + 6455813364 \, x^{3} + 4169655991 \, x^{2} + 11442420 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (5 \, x + 3\right ) - 11442420 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (3 \, x + 2\right ) + 1345680462 \, x + 173603415}{12 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]
1/12*(1544726700*x^5 + 4994616330*x^4 + 6455813364*x^3 + 4169655991*x^2 + 11442420*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log(5*x + 3) - 11442420*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^ 3 + 5224*x^2 + 1344*x + 144)*log(3*x + 2) + 1345680462*x + 173603415)/(202 5*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=- \frac {- 1544726700 x^{5} - 4994616330 x^{4} - 6455813364 x^{3} - 4169655991 x^{2} - 1345680462 x - 173603415}{24300 x^{6} + 93960 x^{5} + 151308 x^{4} + 129888 x^{3} + 62688 x^{2} + 16128 x + 1728} + 953535 \log {\left (x + \frac {3}{5} \right )} - 953535 \log {\left (x + \frac {2}{3} \right )} \]
-(-1544726700*x**5 - 4994616330*x**4 - 6455813364*x**3 - 4169655991*x**2 - 1345680462*x - 173603415)/(24300*x**6 + 93960*x**5 + 151308*x**4 + 129888 *x**3 + 62688*x**2 + 16128*x + 1728) + 953535*log(x + 3/5) - 953535*log(x + 2/3)
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {1544726700 \, x^{5} + 4994616330 \, x^{4} + 6455813364 \, x^{3} + 4169655991 \, x^{2} + 1345680462 \, x + 173603415}{12 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} + 953535 \, \log \left (5 \, x + 3\right ) - 953535 \, \log \left (3 \, x + 2\right ) \]
1/12*(1544726700*x^5 + 4994616330*x^4 + 6455813364*x^3 + 4169655991*x^2 + 1345680462*x + 173603415)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5 224*x^2 + 1344*x + 144) + 953535*log(5*x + 3) - 953535*log(3*x + 2)
Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {128634}{3 \, x + 2} - \frac {27225 \, {\left (\frac {136}{3 \, x + 2} - 625\right )}}{2 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {7854}{{\left (3 \, x + 2\right )}^{2}} + \frac {539}{{\left (3 \, x + 2\right )}^{3}} + \frac {343}{12 \, {\left (3 \, x + 2\right )}^{4}} + 953535 \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) \]
128634/(3*x + 2) - 27225/2*(136/(3*x + 2) - 625)/(1/(3*x + 2) - 5)^2 + 785 4/(3*x + 2)^2 + 539/(3*x + 2)^3 + 343/12/(3*x + 2)^4 + 953535*log(abs(-1/( 3*x + 2) + 5))
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {63569\,x^5+\frac {6166193\,x^4}{30}+\frac {179328149\,x^3}{675}+\frac {4169655991\,x^2}{24300}+\frac {224280077\,x}{4050}+\frac {11573561}{1620}}{x^6+\frac {58\,x^5}{15}+\frac {467\,x^4}{75}+\frac {3608\,x^3}{675}+\frac {5224\,x^2}{2025}+\frac {448\,x}{675}+\frac {16}{225}}-1907070\,\mathrm {atanh}\left (30\,x+19\right ) \]