Integrand size = 20, antiderivative size = 86 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {21}{5 (2+3 x)^5}+\frac {309}{4 (2+3 x)^4}+\frac {1020}{(2+3 x)^3}+\frac {12675}{(2+3 x)^2}+\frac {189375}{2+3 x}-\frac {6875}{2 (3+5 x)^2}+\frac {125000}{3+5 x}-1321875 \log (2+3 x)+1321875 \log (3+5 x) \]
21/5/(2+3*x)^5+309/4/(2+3*x)^4+1020/(2+3*x)^3+12675/(2+3*x)^2+189375/(2+3* x)-6875/2/(3+5*x)^2+125000/(3+5*x)-1321875*ln(2+3*x)+1321875*ln(3+5*x)
Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {21}{5 (2+3 x)^5}+\frac {309}{4 (2+3 x)^4}+\frac {1020}{(2+3 x)^3}+\frac {12675}{(2+3 x)^2}+\frac {189375}{2+3 x}-\frac {6875}{2 (3+5 x)^2}+\frac {125000}{3+5 x}-1321875 \log (2+3 x)+1321875 \log (-3 (3+5 x)) \]
21/(5*(2 + 3*x)^5) + 309/(4*(2 + 3*x)^4) + 1020/(2 + 3*x)^3 + 12675/(2 + 3 *x)^2 + 189375/(2 + 3*x) - 6875/(2*(3 + 5*x)^2) + 125000/(3 + 5*x) - 13218 75*Log[2 + 3*x] + 1321875*Log[-3*(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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.
\(\displaystyle \int \frac {1-2 x}{(3 x+2)^6 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {6609375}{5 x+3}-\frac {625000}{(5 x+3)^2}+\frac {34375}{(5 x+3)^3}-\frac {3965625}{3 x+2}-\frac {568125}{(3 x+2)^2}-\frac {76050}{(3 x+2)^3}-\frac {9180}{(3 x+2)^4}-\frac {927}{(3 x+2)^5}-\frac {63}{(3 x+2)^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {189375}{3 x+2}+\frac {125000}{5 x+3}+\frac {12675}{(3 x+2)^2}-\frac {6875}{2 (5 x+3)^2}+\frac {1020}{(3 x+2)^3}+\frac {309}{4 (3 x+2)^4}+\frac {21}{5 (3 x+2)^5}-1321875 \log (3 x+2)+1321875 \log (5 x+3)\) |
21/(5*(2 + 3*x)^5) + 309/(4*(2 + 3*x)^4) + 1020/(2 + 3*x)^3 + 12675/(2 + 3 *x)^2 + 189375/(2 + 3*x) - 6875/(2*(3 + 5*x)^2) + 125000/(3 + 5*x) - 13218 75*Log[2 + 3*x] + 1321875*Log[3 + 5*x]
3.13.35.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Time = 2.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.73
method | result | size |
norman | \(\frac {535359375 x^{6}+1429766790 x^{2}+3391402500 x^{4}+\frac {1484332427}{4} x +\frac {4175803125}{2} x^{5}+\frac {11746762875}{4} x^{3}+\frac {401107483}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-1321875 \ln \left (2+3 x \right )+1321875 \ln \left (3+5 x \right )\) | \(63\) |
risch | \(\frac {535359375 x^{6}+1429766790 x^{2}+3391402500 x^{4}+\frac {1484332427}{4} x +\frac {4175803125}{2} x^{5}+\frac {11746762875}{4} x^{3}+\frac {401107483}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-1321875 \ln \left (2+3 x \right )+1321875 \ln \left (3+5 x \right )\) | \(64\) |
default | \(\frac {21}{5 \left (2+3 x \right )^{5}}+\frac {309}{4 \left (2+3 x \right )^{4}}+\frac {1020}{\left (2+3 x \right )^{3}}+\frac {12675}{\left (2+3 x \right )^{2}}+\frac {189375}{2+3 x}-\frac {6875}{2 \left (3+5 x \right )^{2}}+\frac {125000}{3+5 x}-1321875 \ln \left (2+3 x \right )+1321875 \ln \left (3+5 x \right )\) | \(81\) |
parallelrisch | \(-\frac {182735999520 x -55125360000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+142077240000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-11877840000000 \ln \left (x +\frac {3}{5}\right ) x +55125360000000 \ln \left (\frac {2}{3}+x \right ) x^{2}+11877840000000 \ln \left (\frac {2}{3}+x \right ) x +15440879443221 x^{5}+9504665081820 x^{6}+2436727959225 x^{7}+6511661995560 x^{3}+13372651494270 x^{4}+1690307998640 x^{2}+219625830000000 \ln \left (\frac {2}{3}+x \right ) x^{4}+1096416000000 \ln \left (\frac {2}{3}+x \right )+23127525000000 \ln \left (\frac {2}{3}+x \right ) x^{7}-23127525000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-1096416000000 \ln \left (x +\frac {3}{5}\right )+203625009000000 \ln \left (\frac {2}{3}+x \right ) x^{5}-142077240000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-203625009000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-219625830000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+104844780000000 \ln \left (\frac {2}{3}+x \right ) x^{6}-104844780000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{2880 \left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}\) | \(185\) |
(535359375*x^6+1429766790*x^2+3391402500*x^4+1484332427/4*x+4175803125/2*x ^5+11746762875/4*x^3+401107483/10)/(2+3*x)^5/(3+5*x)^2-1321875*ln(2+3*x)+1 321875*ln(3+5*x)
Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.80 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {10707187500 \, x^{6} + 41758031250 \, x^{5} + 67828050000 \, x^{4} + 58733814375 \, x^{3} + 28595335800 \, x^{2} + 26437500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (5 \, x + 3\right ) - 26437500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (3 \, x + 2\right ) + 7421662135 \, x + 802214966}{20 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \]
1/20*(10707187500*x^6 + 41758031250*x^5 + 67828050000*x^4 + 58733814375*x^ 3 + 28595335800*x^2 + 26437500*(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) - 26437500*(6075*x ^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)*log(3*x + 2) + 7421662135*x + 802214966)/(6075*x^7 + 27540*x^6 + 5348 7*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 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=- \frac {- 10707187500 x^{6} - 41758031250 x^{5} - 67828050000 x^{4} - 58733814375 x^{3} - 28595335800 x^{2} - 7421662135 x - 802214966}{121500 x^{7} + 550800 x^{6} + 1069740 x^{5} + 1153800 x^{4} + 746400 x^{3} + 289600 x^{2} + 62400 x + 5760} + 1321875 \log {\left (x + \frac {3}{5} \right )} - 1321875 \log {\left (x + \frac {2}{3} \right )} \]
-(-10707187500*x**6 - 41758031250*x**5 - 67828050000*x**4 - 58733814375*x* *3 - 28595335800*x**2 - 7421662135*x - 802214966)/(121500*x**7 + 550800*x* *6 + 1069740*x**5 + 1153800*x**4 + 746400*x**3 + 289600*x**2 + 62400*x + 5 760) + 1321875*log(x + 3/5) - 1321875*log(x + 2/3)
Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {10707187500 \, x^{6} + 41758031250 \, x^{5} + 67828050000 \, x^{4} + 58733814375 \, x^{3} + 28595335800 \, x^{2} + 7421662135 \, x + 802214966}{20 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} + 1321875 \, \log \left (5 \, x + 3\right ) - 1321875 \, \log \left (3 \, x + 2\right ) \]
1/20*(10707187500*x^6 + 41758031250*x^5 + 67828050000*x^4 + 58733814375*x^ 3 + 28595335800*x^2 + 7421662135*x + 802214966)/(6075*x^7 + 27540*x^6 + 53 487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288) + 1321875*log( 5*x + 3) - 1321875*log(3*x + 2)
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {10707187500 \, x^{6} + 41758031250 \, x^{5} + 67828050000 \, x^{4} + 58733814375 \, x^{3} + 28595335800 \, x^{2} + 7421662135 \, x + 802214966}{20 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{5}} + 1321875 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 1321875 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]
1/20*(10707187500*x^6 + 41758031250*x^5 + 67828050000*x^4 + 58733814375*x^ 3 + 28595335800*x^2 + 7421662135*x + 802214966)/((5*x + 3)^2*(3*x + 2)^5) + 1321875*log(abs(5*x + 3)) - 1321875*log(abs(3*x + 2))
Time = 1.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {88125\,x^6+\frac {687375\,x^5}{2}+\frac {5024300\,x^4}{9}+\frac {52207835\,x^3}{108}+\frac {95317786\,x^2}{405}+\frac {1484332427\,x}{24300}+\frac {401107483}{60750}}{x^7+\frac {68\,x^6}{15}+\frac {1981\,x^5}{225}+\frac {1282\,x^4}{135}+\frac {2488\,x^3}{405}+\frac {2896\,x^2}{1215}+\frac {208\,x}{405}+\frac {32}{675}}-2643750\,\mathrm {atanh}\left (30\,x+19\right ) \]