Integrand size = 22, antiderivative size = 86 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {343}{15 (2+3 x)^5}+\frac {1617}{4 (2+3 x)^4}+\frac {5236}{(2+3 x)^3}+\frac {64317}{(2+3 x)^2}+\frac {953535}{2+3 x}-\frac {33275}{2 (3+5 x)^2}+\frac {617100}{3+5 x}-6618975 \log (2+3 x)+6618975 \log (3+5 x) \]
343/15/(2+3*x)^5+1617/4/(2+3*x)^4+5236/(2+3*x)^3+64317/(2+3*x)^2+953535/(2 +3*x)-33275/2/(3+5*x)^2+617100/(3+5*x)-6618975*ln(2+3*x)+6618975*ln(3+5*x)
Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {343}{15 (2+3 x)^5}+\frac {1617}{4 (2+3 x)^4}+\frac {5236}{(2+3 x)^3}+\frac {64317}{(2+3 x)^2}+\frac {953535}{2+3 x}-\frac {33275}{2 (3+5 x)^2}+\frac {617100}{3+5 x}-6618975 \log (5 (2+3 x))+6618975 \log (3+5 x) \]
343/(15*(2 + 3*x)^5) + 1617/(4*(2 + 3*x)^4) + 5236/(2 + 3*x)^3 + 64317/(2 + 3*x)^2 + 953535/(2 + 3*x) - 33275/(2*(3 + 5*x)^2) + 617100/(3 + 5*x) - 6 618975*Log[5*(2 + 3*x)] + 6618975*Log[3 + 5*x]
Time = 0.22 (sec) , antiderivative size = 86, 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.
\(\displaystyle \int \frac {(1-2 x)^3}{(3 x+2)^6 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {33094875}{5 x+3}-\frac {3085500}{(5 x+3)^2}+\frac {166375}{(5 x+3)^3}-\frac {19856925}{3 x+2}-\frac {2860605}{(3 x+2)^2}-\frac {385902}{(3 x+2)^3}-\frac {47124}{(3 x+2)^4}-\frac {4851}{(3 x+2)^5}-\frac {343}{(3 x+2)^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {953535}{3 x+2}+\frac {617100}{5 x+3}+\frac {64317}{(3 x+2)^2}-\frac {33275}{2 (5 x+3)^2}+\frac {5236}{(3 x+2)^3}+\frac {1617}{4 (3 x+2)^4}+\frac {343}{15 (3 x+2)^5}-6618975 \log (3 x+2)+6618975 \log (5 x+3)\) |
343/(15*(2 + 3*x)^5) + 1617/(4*(2 + 3*x)^4) + 5236/(2 + 3*x)^3 + 64317/(2 + 3*x)^2 + 953535/(2 + 3*x) - 33275/(2*(3 + 5*x)^2) + 617100/(3 + 5*x) - 6 618975*Log[2 + 3*x] + 6618975*Log[3 + 5*x]
3.15.31.3.1 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 = 63, normalized size of antiderivative = 0.73
method | result | size |
norman | \(\frac {2680684875 x^{6}+16981642260 x^{4}+\frac {7432441967}{4} x +\frac {20909342025}{2} x^{5}+\frac {21477652514}{3} x^{2}+\frac {58819124199}{4} x^{3}+\frac {2008450423}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-6618975 \ln \left (2+3 x \right )+6618975 \ln \left (3+5 x \right )\) | \(63\) |
risch | \(\frac {2680684875 x^{6}+16981642260 x^{4}+\frac {7432441967}{4} x +\frac {20909342025}{2} x^{5}+\frac {21477652514}{3} x^{2}+\frac {58819124199}{4} x^{3}+\frac {2008450423}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-6618975 \ln \left (2+3 x \right )+6618975 \ln \left (3+5 x \right )\) | \(64\) |
default | \(\frac {343}{15 \left (2+3 x \right )^{5}}+\frac {1617}{4 \left (2+3 x \right )^{4}}+\frac {5236}{\left (2+3 x \right )^{3}}+\frac {64317}{\left (2+3 x \right )^{2}}+\frac {953535}{2+3 x}-\frac {33275}{2 \left (3+5 x \right )^{2}}+\frac {617100}{3+5 x}-6618975 \ln \left (2+3 x \right )+6618975 \ln \left (3+5 x \right )\) | \(81\) |
parallelrisch | \(-\frac {915007103520 x -276027143040000 \ln \left (x +\frac {3}{5}\right ) x^{2}+711418023360000 \ln \left (\frac {2}{3}+x \right ) x^{3}-59475461760000 \ln \left (x +\frac {3}{5}\right ) x +276027143040000 \ln \left (\frac {2}{3}+x \right ) x^{2}+59475461760000 \ln \left (\frac {2}{3}+x \right ) x +77316535259001 x^{5}+47592352209420 x^{6}+12201336319725 x^{7}+32605600363080 x^{3}+66960375194070 x^{4}+8463815711600 x^{2}+1099724163120000 \ln \left (\frac {2}{3}+x \right ) x^{4}+5490042624000 \ln \left (\frac {2}{3}+x \right )+115805586600000 \ln \left (\frac {2}{3}+x \right ) x^{7}-115805586600000 \ln \left (x +\frac {3}{5}\right ) x^{7}-5490042624000 \ln \left (x +\frac {3}{5}\right )+1019603853576000 \ln \left (\frac {2}{3}+x \right ) x^{5}-711418023360000 \ln \left (x +\frac {3}{5}\right ) x^{3}-1019603853576000 \ln \left (x +\frac {3}{5}\right ) x^{5}-1099724163120000 \ln \left (x +\frac {3}{5}\right ) x^{4}+524985325920000 \ln \left (\frac {2}{3}+x \right ) x^{6}-524985325920000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{2880 \left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}\) | \(185\) |
(2680684875*x^6+16981642260*x^4+7432441967/4*x+20909342025/2*x^5+214776525 14/3*x^2+58819124199/4*x^3+2008450423/10)/(2+3*x)^5/(3+5*x)^2-6618975*ln(2 +3*x)+6618975*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.80 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {160841092500 \, x^{6} + 627280260750 \, x^{5} + 1018898535600 \, x^{4} + 882286862985 \, x^{3} + 429553050280 \, x^{2} + 397138500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (5 \, x + 3\right ) - 397138500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (3 \, x + 2\right ) + 111486629505 \, x + 12050702538}{60 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \]
1/60*(160841092500*x^6 + 627280260750*x^5 + 1018898535600*x^4 + 8822868629 85*x^3 + 429553050280*x^2 + 397138500*(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)*log(5*x + 3) - 397138500 *(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3 120*x + 288)*log(3*x + 2) + 111486629505*x + 12050702538)/(6075*x^7 + 2754 0*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)
Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=- \frac {- 160841092500 x^{6} - 627280260750 x^{5} - 1018898535600 x^{4} - 882286862985 x^{3} - 429553050280 x^{2} - 111486629505 x - 12050702538}{364500 x^{7} + 1652400 x^{6} + 3209220 x^{5} + 3461400 x^{4} + 2239200 x^{3} + 868800 x^{2} + 187200 x + 17280} + 6618975 \log {\left (x + \frac {3}{5} \right )} - 6618975 \log {\left (x + \frac {2}{3} \right )} \]
-(-160841092500*x**6 - 627280260750*x**5 - 1018898535600*x**4 - 8822868629 85*x**3 - 429553050280*x**2 - 111486629505*x - 12050702538)/(364500*x**7 + 1652400*x**6 + 3209220*x**5 + 3461400*x**4 + 2239200*x**3 + 868800*x**2 + 187200*x + 17280) + 6618975*log(x + 3/5) - 6618975*log(x + 2/3)
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {160841092500 \, x^{6} + 627280260750 \, x^{5} + 1018898535600 \, x^{4} + 882286862985 \, x^{3} + 429553050280 \, x^{2} + 111486629505 \, x + 12050702538}{60 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} + 6618975 \, \log \left (5 \, x + 3\right ) - 6618975 \, \log \left (3 \, x + 2\right ) \]
1/60*(160841092500*x^6 + 627280260750*x^5 + 1018898535600*x^4 + 8822868629 85*x^3 + 429553050280*x^2 + 111486629505*x + 12050702538)/(6075*x^7 + 2754 0*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288) + 66 18975*log(5*x + 3) - 6618975*log(3*x + 2)
Time = 0.41 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {160841092500 \, x^{6} + 627280260750 \, x^{5} + 1018898535600 \, x^{4} + 882286862985 \, x^{3} + 429553050280 \, x^{2} + 111486629505 \, x + 12050702538}{60 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{5}} + 6618975 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 6618975 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]
1/60*(160841092500*x^6 + 627280260750*x^5 + 1018898535600*x^4 + 8822868629 85*x^3 + 429553050280*x^2 + 111486629505*x + 12050702538)/((5*x + 3)^2*(3* x + 2)^5) + 6618975*log(abs(5*x + 3)) - 6618975*log(abs(3*x + 2))
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {441265\,x^6+\frac {3441867\,x^5}{2}+\frac {377369828\,x^4}{135}+\frac {19606374733\,x^3}{8100}+\frac {21477652514\,x^2}{18225}+\frac {7432441967\,x}{24300}+\frac {2008450423}{60750}}{x^7+\frac {68\,x^6}{15}+\frac {1981\,x^5}{225}+\frac {1282\,x^4}{135}+\frac {2488\,x^3}{405}+\frac {2896\,x^2}{1215}+\frac {208\,x}{405}+\frac {32}{675}}-13237950\,\mathrm {atanh}\left (30\,x+19\right ) \]