Integrand size = 20, antiderivative size = 90 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {7}{6 (2+3 x)^6}-\frac {68}{5 (2+3 x)^5}-\frac {505}{4 (2+3 x)^4}-\frac {3350}{3 (2+3 x)^3}-\frac {20875}{2 (2+3 x)^2}-\frac {125000}{2+3 x}-\frac {34375}{3+5 x}+728125 \log (2+3 x)-728125 \log (3+5 x) \]
-7/6/(2+3*x)^6-68/5/(2+3*x)^5-505/4/(2+3*x)^4-3350/3/(2+3*x)^3-20875/2/(2+ 3*x)^2-125000/(2+3*x)-34375/(3+5*x)+728125*ln(2+3*x)-728125*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {7}{6 (2+3 x)^6}-\frac {68}{5 (2+3 x)^5}-\frac {505}{4 (2+3 x)^4}-\frac {3350}{3 (2+3 x)^3}-\frac {20875}{2 (2+3 x)^2}-\frac {125000}{2+3 x}-\frac {34375}{3+5 x}+728125 \log (2+3 x)-728125 \log (-3 (3+5 x)) \]
-7/(6*(2 + 3*x)^6) - 68/(5*(2 + 3*x)^5) - 505/(4*(2 + 3*x)^4) - 3350/(3*(2 + 3*x)^3) - 20875/(2*(2 + 3*x)^2) - 125000/(2 + 3*x) - 34375/(3 + 5*x) + 728125*Log[2 + 3*x] - 728125*Log[-3*(3 + 5*x)]
Time = 0.22 (sec) , antiderivative size = 90, 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.
\(\displaystyle \int \frac {1-2 x}{(3 x+2)^7 (5 x+3)^2} \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (-\frac {3640625}{5 x+3}+\frac {171875}{(5 x+3)^2}+\frac {2184375}{3 x+2}+\frac {375000}{(3 x+2)^2}+\frac {62625}{(3 x+2)^3}+\frac {10050}{(3 x+2)^4}+\frac {1515}{(3 x+2)^5}+\frac {204}{(3 x+2)^6}+\frac {21}{(3 x+2)^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {125000}{3 x+2}-\frac {34375}{5 x+3}-\frac {20875}{2 (3 x+2)^2}-\frac {3350}{3 (3 x+2)^3}-\frac {505}{4 (3 x+2)^4}-\frac {68}{5 (3 x+2)^5}-\frac {7}{6 (3 x+2)^6}+728125 \log (3 x+2)-728125 \log (5 x+3)\) |
-7/(6*(2 + 3*x)^6) - 68/(5*(2 + 3*x)^5) - 505/(4*(2 + 3*x)^4) - 3350/(3*(2 + 3*x)^3) - 20875/(2*(2 + 3*x)^2) - 125000/(2 + 3*x) - 34375/(3 + 5*x) + 728125*Log[2 + 3*x] - 728125*Log[3 + 5*x]
3.13.21.3.1 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 = 2.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.70
method | result | size |
norman | \(\frac {-176934375 x^{6}-\frac {4087734525}{4} x^{3}-\frac {2319544125}{2} x^{4}-\frac {2025666351}{4} x^{2}-\frac {1403679375}{2} x^{5}-\frac {1338136009}{10} x -\frac {147294001}{10}}{\left (2+3 x \right )^{6} \left (3+5 x \right )}+728125 \ln \left (2+3 x \right )-728125 \ln \left (3+5 x \right )\) | \(63\) |
risch | \(\frac {-176934375 x^{6}-\frac {4087734525}{4} x^{3}-\frac {2319544125}{2} x^{4}-\frac {2025666351}{4} x^{2}-\frac {1403679375}{2} x^{5}-\frac {1338136009}{10} x -\frac {147294001}{10}}{\left (2+3 x \right )^{6} \left (3+5 x \right )}+728125 \ln \left (2+3 x \right )-728125 \ln \left (3+5 x \right )\) | \(64\) |
default | \(-\frac {7}{6 \left (2+3 x \right )^{6}}-\frac {68}{5 \left (2+3 x \right )^{5}}-\frac {505}{4 \left (2+3 x \right )^{4}}-\frac {3350}{3 \left (2+3 x \right )^{3}}-\frac {20875}{2 \left (2+3 x \right )^{2}}-\frac {125000}{2+3 x}-\frac {34375}{3+5 x}+728125 \ln \left (2+3 x \right )-728125 \ln \left (3+5 x \right )\) | \(81\) |
parallelrisch | \(\frac {44736000320 x -13085280000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+33216480000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-2863104000000 \ln \left (x +\frac {3}{5}\right ) x +13085280000000 \ln \left (\frac {2}{3}+x \right ) x^{2}+2863104000000 \ln \left (\frac {2}{3}+x \right ) x +3520239945048 x^{5}+2129964514767 x^{6}+536886633645 x^{7}+1537592891760 x^{3}+3102334596180 x^{4}+406352000880 x^{2}+50579640000000 \ln \left (\frac {2}{3}+x \right ) x^{4}+268416000000 \ln \left (\frac {2}{3}+x \right )+5095710000000 \ln \left (\frac {2}{3}+x \right ) x^{7}-5095710000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-268416000000 \ln \left (x +\frac {3}{5}\right )+46201104000000 \ln \left (\frac {2}{3}+x \right ) x^{5}-33216480000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-46201104000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-50579640000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+23440266000000 \ln \left (\frac {2}{3}+x \right ) x^{6}-23440266000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{1920 \left (2+3 x \right )^{6} \left (3+5 x \right )}\) | \(185\) |
(-176934375*x^6-4087734525/4*x^3-2319544125/2*x^4-2025666351/4*x^2-1403679 375/2*x^5-1338136009/10*x-147294001/10)/(2+3*x)^6/(3+5*x)+728125*ln(2+3*x) -728125*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.72 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {3538687500 \, x^{6} + 14036793750 \, x^{5} + 23195441250 \, x^{4} + 20438672625 \, x^{3} + 10128331755 \, x^{2} + 14562500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (5 \, x + 3\right ) - 14562500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (3 \, x + 2\right ) + 2676272018 \, x + 294588002}{20 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} \]
-1/20*(3538687500*x^6 + 14036793750*x^5 + 23195441250*x^4 + 20438672625*x^ 3 + 10128331755*x^2 + 14562500*(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x ^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(5*x + 3) - 14562500*(3645*x^ 7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 19 2)*log(3*x + 2) + 2676272018*x + 294588002)/(3645*x^7 + 16767*x^6 + 33048* x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.91 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=- \frac {3538687500 x^{6} + 14036793750 x^{5} + 23195441250 x^{4} + 20438672625 x^{3} + 10128331755 x^{2} + 2676272018 x + 294588002}{72900 x^{7} + 335340 x^{6} + 660960 x^{5} + 723600 x^{4} + 475200 x^{3} + 187200 x^{2} + 40960 x + 3840} - 728125 \log {\left (x + \frac {3}{5} \right )} + 728125 \log {\left (x + \frac {2}{3} \right )} \]
-(3538687500*x**6 + 14036793750*x**5 + 23195441250*x**4 + 20438672625*x**3 + 10128331755*x**2 + 2676272018*x + 294588002)/(72900*x**7 + 335340*x**6 + 660960*x**5 + 723600*x**4 + 475200*x**3 + 187200*x**2 + 40960*x + 3840) - 728125*log(x + 3/5) + 728125*log(x + 2/3)
Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {3538687500 \, x^{6} + 14036793750 \, x^{5} + 23195441250 \, x^{4} + 20438672625 \, x^{3} + 10128331755 \, x^{2} + 2676272018 \, x + 294588002}{20 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} - 728125 \, \log \left (5 \, x + 3\right ) + 728125 \, \log \left (3 \, x + 2\right ) \]
-1/20*(3538687500*x^6 + 14036793750*x^5 + 23195441250*x^4 + 20438672625*x^ 3 + 10128331755*x^2 + 2676272018*x + 294588002)/(3645*x^7 + 16767*x^6 + 33 048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192) - 728125*log(5* x + 3) + 728125*log(3*x + 2)
Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {34375}{5 \, x + 3} + \frac {5625 \, {\left (\frac {1100034}{5 \, x + 3} + \frac {811665}{{\left (5 \, x + 3\right )}^{2}} + \frac {304700}{{\left (5 \, x + 3\right )}^{3}} + \frac {58650}{{\left (5 \, x + 3\right )}^{4}} + \frac {4700}{{\left (5 \, x + 3\right )}^{5}} + 604017\right )}}{4 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{6}} + 728125 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]
-34375/(5*x + 3) + 5625/4*(1100034/(5*x + 3) + 811665/(5*x + 3)^2 + 304700 /(5*x + 3)^3 + 58650/(5*x + 3)^4 + 4700/(5*x + 3)^5 + 604017)/(1/(5*x + 3) + 3)^6 + 728125*log(abs(-1/(5*x + 3) - 3))
Time = 1.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=1456250\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {145625\,x^6}{3}+\frac {3465875\,x^5}{18}+\frac {51545425\,x^4}{162}+\frac {30279515\,x^3}{108}+\frac {225074039\,x^2}{1620}+\frac {1338136009\,x}{36450}+\frac {147294001}{36450}}{x^7+\frac {23\,x^6}{5}+\frac {136\,x^5}{15}+\frac {268\,x^4}{27}+\frac {176\,x^3}{27}+\frac {208\,x^2}{81}+\frac {2048\,x}{3645}+\frac {64}{1215}} \]