Integrand size = 20, antiderivative size = 81 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {7}{18 (2+3 x)^6}+\frac {11}{5 (2+3 x)^5}+\frac {55}{4 (2+3 x)^4}+\frac {275}{3 (2+3 x)^3}+\frac {1375}{2 (2+3 x)^2}+\frac {6875}{2+3 x}-34375 \log (2+3 x)+34375 \log (3+5 x) \]
7/18/(2+3*x)^6+11/5/(2+3*x)^5+55/4/(2+3*x)^4+275/3/(2+3*x)^3+1375/2/(2+3*x )^2+6875/(2+3*x)-34375*ln(2+3*x)+34375*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {70+396 (2+3 x)+2475 (2+3 x)^2+16500 (2+3 x)^3+123750 (2+3 x)^4+1237500 (2+3 x)^5}{180 (2+3 x)^6}-34375 \log (2+3 x)+34375 \log (-3 (3+5 x)) \]
(70 + 396*(2 + 3*x) + 2475*(2 + 3*x)^2 + 16500*(2 + 3*x)^3 + 123750*(2 + 3 *x)^4 + 1237500*(2 + 3*x)^5)/(180*(2 + 3*x)^6) - 34375*Log[2 + 3*x] + 3437 5*Log[-3*(3 + 5*x)]
Time = 0.20 (sec) , antiderivative size = 81, 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.
\(\displaystyle \int \frac {1-2 x}{(3 x+2)^7 (5 x+3)} \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {171875}{5 x+3}-\frac {103125}{3 x+2}-\frac {20625}{(3 x+2)^2}-\frac {4125}{(3 x+2)^3}-\frac {825}{(3 x+2)^4}-\frac {165}{(3 x+2)^5}-\frac {33}{(3 x+2)^6}-\frac {7}{(3 x+2)^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6875}{3 x+2}+\frac {1375}{2 (3 x+2)^2}+\frac {275}{3 (3 x+2)^3}+\frac {55}{4 (3 x+2)^4}+\frac {11}{5 (3 x+2)^5}+\frac {7}{18 (3 x+2)^6}-34375 \log (3 x+2)+34375 \log (5 x+3)\) |
7/(18*(2 + 3*x)^6) + 11/(5*(2 + 3*x)^5) + 55/(4*(2 + 3*x)^4) + 275/(3*(2 + 3*x)^3) + 1375/(2*(2 + 3*x)^2) + 6875/(2 + 3*x) - 34375*Log[2 + 3*x] + 34 375*Log[3 + 5*x]
3.13.6.3.1 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 = 51, normalized size of antiderivative = 0.63
method | result | size |
norman | \(\frac {1670625 x^{5}+7575975 x^{3}+\frac {8597358}{5} x +\frac {11248875}{2} x^{4}+\frac {20414295}{4} x^{2}+\frac {20861381}{90}}{\left (2+3 x \right )^{6}}-34375 \ln \left (2+3 x \right )+34375 \ln \left (3+5 x \right )\) | \(51\) |
risch | \(\frac {1670625 x^{5}+7575975 x^{3}+\frac {8597358}{5} x +\frac {11248875}{2} x^{4}+\frac {20414295}{4} x^{2}+\frac {20861381}{90}}{\left (2+3 x \right )^{6}}-34375 \ln \left (2+3 x \right )+34375 \ln \left (3+5 x \right )\) | \(52\) |
default | \(\frac {7}{18 \left (2+3 x \right )^{6}}+\frac {11}{5 \left (2+3 x \right )^{5}}+\frac {55}{4 \left (2+3 x \right )^{4}}+\frac {275}{3 \left (2+3 x \right )^{3}}+\frac {1375}{2 \left (2+3 x \right )^{2}}+\frac {6875}{2+3 x}-34375 \ln \left (2+3 x \right )+34375 \ln \left (3+5 x \right )\) | \(72\) |
parallelrisch | \(-\frac {234666560 x -47520000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+95040000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-12672000000 \ln \left (x +\frac {3}{5}\right ) x +47520000000 \ln \left (\frac {2}{3}+x \right ) x^{2}+12672000000 \ln \left (\frac {2}{3}+x \right ) x +5689887444 x^{5}+1689771861 x^{6}+5164838880 x^{3}+7665505740 x^{4}+1740444240 x^{2}+106920000000 \ln \left (\frac {2}{3}+x \right ) x^{4}+1408000000 \ln \left (\frac {2}{3}+x \right )-1408000000 \ln \left (x +\frac {3}{5}\right )+64152000000 \ln \left (\frac {2}{3}+x \right ) x^{5}-95040000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-64152000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-106920000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+16038000000 \ln \left (\frac {2}{3}+x \right ) x^{6}-16038000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{640 \left (2+3 x \right )^{6}}\) | \(155\) |
(1670625*x^5+7575975*x^3+8597358/5*x+11248875/2*x^4+20414295/4*x^2+2086138 1/90)/(2+3*x)^6-34375*ln(2+3*x)+34375*ln(3+5*x)
Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.67 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 6187500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (5 \, x + 3\right ) - 6187500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 309504888 \, x + 41722762}{180 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 6 187500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)* log(5*x + 3) - 6187500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^ 2 + 576*x + 64)*log(3*x + 2) + 309504888*x + 41722762)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=- \frac {- 300712500 x^{5} - 1012398750 x^{4} - 1363675500 x^{3} - 918643275 x^{2} - 309504888 x - 41722762}{131220 x^{6} + 524880 x^{5} + 874800 x^{4} + 777600 x^{3} + 388800 x^{2} + 103680 x + 11520} + 34375 \log {\left (x + \frac {3}{5} \right )} - 34375 \log {\left (x + \frac {2}{3} \right )} \]
-(-300712500*x**5 - 1012398750*x**4 - 1363675500*x**3 - 918643275*x**2 - 3 09504888*x - 41722762)/(131220*x**6 + 524880*x**5 + 874800*x**4 + 777600*x **3 + 388800*x**2 + 103680*x + 11520) + 34375*log(x + 3/5) - 34375*log(x + 2/3)
Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 309504888 \, x + 41722762}{180 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + 34375 \, \log \left (5 \, x + 3\right ) - 34375 \, \log \left (3 \, x + 2\right ) \]
1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 3 09504888*x + 41722762)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^ 2 + 576*x + 64) + 34375*log(5*x + 3) - 34375*log(3*x + 2)
Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.65 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 309504888 \, x + 41722762}{180 \, {\left (3 \, x + 2\right )}^{6}} + 34375 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 34375 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]
1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 3 09504888*x + 41722762)/(3*x + 2)^6 + 34375*log(abs(5*x + 3)) - 34375*log(a bs(3*x + 2))
Time = 1.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.80 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {\frac {6875\,x^5}{3}+\frac {138875\,x^4}{18}+\frac {841775\,x^3}{81}+\frac {756085\,x^2}{108}+\frac {955262\,x}{405}+\frac {20861381}{65610}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}}-68750\,\mathrm {atanh}\left (30\,x+19\right ) \]