Integrand size = 20, antiderivative size = 66 \[ \int \frac {1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {7}{(2+3 x)^3}+\frac {309}{2 (2+3 x)^2}+\frac {3060}{2+3 x}-\frac {275}{2 (3+5 x)^2}+\frac {3350}{3+5 x}-25350 \log (2+3 x)+25350 \log (3+5 x) \] Output:
7/(2+3*x)^3+309/2/(2+3*x)^2+3060/(2+3*x)-275/2/(3+5*x)^2+3350/(3+5*x)-2535 0*ln(2+3*x)+25350*ln(3+5*x)
Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \frac {1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {7}{(2+3 x)^3}+\frac {309}{2 (2+3 x)^2}+\frac {3060}{2+3 x}-\frac {275}{2 (3+5 x)^2}+\frac {3350}{3+5 x}-25350 \log (2+3 x)+25350 \log (-3 (3+5 x)) \] Input:
Integrate[(1 - 2*x)/((2 + 3*x)^4*(3 + 5*x)^3),x]
Output:
7/(2 + 3*x)^3 + 309/(2*(2 + 3*x)^2) + 3060/(2 + 3*x) - 275/(2*(3 + 5*x)^2) + 3350/(3 + 5*x) - 25350*Log[2 + 3*x] + 25350*Log[-3*(3 + 5*x)]
Time = 0.23 (sec) , antiderivative size = 66, 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.
\(\displaystyle \int \frac {1-2 x}{(3 x+2)^4 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {126750}{5 x+3}-\frac {16750}{(5 x+3)^2}+\frac {1375}{(5 x+3)^3}-\frac {76050}{3 x+2}-\frac {9180}{(3 x+2)^2}-\frac {927}{(3 x+2)^3}-\frac {63}{(3 x+2)^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3060}{3 x+2}+\frac {3350}{5 x+3}+\frac {309}{2 (3 x+2)^2}-\frac {275}{2 (5 x+3)^2}+\frac {7}{(3 x+2)^3}-25350 \log (3 x+2)+25350 \log (5 x+3)\) |
Input:
Int[(1 - 2*x)/((2 + 3*x)^4*(3 + 5*x)^3),x]
Output:
7/(2 + 3*x)^3 + 309/(2*(2 + 3*x)^2) + 3060/(2 + 3*x) - 275/(2*(3 + 5*x)^2) + 3350/(3 + 5*x) - 25350*Log[2 + 3*x] + 25350*Log[3 + 5*x]
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 = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80
method | result | size |
norman | \(\frac {1140750 x^{4}+2815540 x^{2}+2927925 x^{3}+\frac {2404363}{2} x +192304}{\left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-25350 \ln \left (2+3 x \right )+25350 \ln \left (3+5 x \right )\) | \(53\) |
risch | \(\frac {1140750 x^{4}+2815540 x^{2}+2927925 x^{3}+\frac {2404363}{2} x +192304}{\left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-25350 \ln \left (2+3 x \right )+25350 \ln \left (3+5 x \right )\) | \(54\) |
default | \(\frac {7}{\left (2+3 x \right )^{3}}+\frac {309}{2 \left (2+3 x \right )^{2}}+\frac {3060}{2+3 x}-\frac {275}{2 \left (3+5 x \right )^{2}}+\frac {3350}{3+5 x}-25350 \ln \left (2+3 x \right )+25350 \ln \left (3+5 x \right )\) | \(63\) |
parallelrisch | \(-\frac {1232010000 \ln \left (\frac {2}{3}+x \right ) x^{5}-1232010000 \ln \left (x +\frac {3}{5}\right ) x^{5}+3942432000 \ln \left (\frac {2}{3}+x \right ) x^{4}-3942432000 \ln \left (x +\frac {3}{5}\right ) x^{4}+129805200 x^{5}+5043027600 \ln \left (\frac {2}{3}+x \right ) x^{3}-5043027600 \ln \left (x +\frac {3}{5}\right ) x^{3}+333242640 x^{4}+3223303200 \ln \left (\frac {2}{3}+x \right ) x^{2}-3223303200 \ln \left (x +\frac {3}{5}\right ) x^{2}+320525352 x^{3}+1029412800 \ln \left (\frac {2}{3}+x \right ) x -1029412800 \ln \left (x +\frac {3}{5}\right ) x +136889984 x^{2}+131414400 \ln \left (\frac {2}{3}+x \right )-131414400 \ln \left (x +\frac {3}{5}\right )+21902388 x}{72 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}\) | \(139\) |
Input:
int((1-2*x)/(2+3*x)^4/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
(1140750*x^4+2815540*x^2+2927925*x^3+2404363/2*x+192304)/(2+3*x)^3/(3+5*x) ^2-25350*ln(2+3*x)+25350*ln(3+5*x)
Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.74 \[ \int \frac {1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {2281500 \, x^{4} + 5855850 \, x^{3} + 5631080 \, x^{2} + 50700 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 50700 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (3 \, x + 2\right ) + 2404363 \, x + 384608}{2 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \] Input:
integrate((1-2*x)/(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")
Output:
1/2*(2281500*x^4 + 5855850*x^3 + 5631080*x^2 + 50700*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(5*x + 3) - 50700*(675*x^5 + 2160*x^ 4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(3*x + 2) + 2404363*x + 384608)/( 675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int \frac {1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx=- \frac {- 2281500 x^{4} - 5855850 x^{3} - 5631080 x^{2} - 2404363 x - 384608}{1350 x^{5} + 4320 x^{4} + 5526 x^{3} + 3532 x^{2} + 1128 x + 144} + 25350 \log {\left (x + \frac {3}{5} \right )} - 25350 \log {\left (x + \frac {2}{3} \right )} \] Input:
integrate((1-2*x)/(2+3*x)**4/(3+5*x)**3,x)
Output:
-(-2281500*x**4 - 5855850*x**3 - 5631080*x**2 - 2404363*x - 384608)/(1350* x**5 + 4320*x**4 + 5526*x**3 + 3532*x**2 + 1128*x + 144) + 25350*log(x + 3 /5) - 25350*log(x + 2/3)
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {2281500 \, x^{4} + 5855850 \, x^{3} + 5631080 \, x^{2} + 2404363 \, x + 384608}{2 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} + 25350 \, \log \left (5 \, x + 3\right ) - 25350 \, \log \left (3 \, x + 2\right ) \] Input:
integrate((1-2*x)/(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")
Output:
1/2*(2281500*x^4 + 5855850*x^3 + 5631080*x^2 + 2404363*x + 384608)/(675*x^ 5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72) + 25350*log(5*x + 3) - 25 350*log(3*x + 2)
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83 \[ \int \frac {1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {2281500 \, x^{4} + 5855850 \, x^{3} + 5631080 \, x^{2} + 2404363 \, x + 384608}{2 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{3}} + 25350 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 25350 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \] Input:
integrate((1-2*x)/(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")
Output:
1/2*(2281500*x^4 + 5855850*x^3 + 5631080*x^2 + 2404363*x + 384608)/((5*x + 3)^2*(3*x + 2)^3) + 25350*log(abs(5*x + 3)) - 25350*log(abs(3*x + 2))
Time = 1.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83 \[ \int \frac {1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {1690\,x^4+\frac {13013\,x^3}{3}+\frac {563108\,x^2}{135}+\frac {2404363\,x}{1350}+\frac {192304}{675}}{x^5+\frac {16\,x^4}{5}+\frac {307\,x^3}{75}+\frac {1766\,x^2}{675}+\frac {188\,x}{225}+\frac {8}{75}}-50700\,\mathrm {atanh}\left (30\,x+19\right ) \] Input:
int(-(2*x - 1)/((3*x + 2)^4*(5*x + 3)^3),x)
Output:
((2404363*x)/1350 + (563108*x^2)/135 + (13013*x^3)/3 + 1690*x^4 + 192304/6 75)/((188*x)/225 + (1766*x^2)/675 + (307*x^3)/75 + (16*x^4)/5 + x^5 + 8/75 ) - 50700*atanh(30*x + 19)
Time = 0.15 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.58 \[ \int \frac {1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {136890000 \,\mathrm {log}\left (5 x +3\right ) x^{5}+438048000 \,\mathrm {log}\left (5 x +3\right ) x^{4}+560336400 \,\mathrm {log}\left (5 x +3\right ) x^{3}+358144800 \,\mathrm {log}\left (5 x +3\right ) x^{2}+114379200 \,\mathrm {log}\left (5 x +3\right ) x +14601600 \,\mathrm {log}\left (5 x +3\right )-136890000 \,\mathrm {log}\left (3 x +2\right ) x^{5}-438048000 \,\mathrm {log}\left (3 x +2\right ) x^{4}-560336400 \,\mathrm {log}\left (3 x +2\right ) x^{3}-358144800 \,\mathrm {log}\left (3 x +2\right ) x^{2}-114379200 \,\mathrm {log}\left (3 x +2\right ) x -14601600 \,\mathrm {log}\left (3 x +2\right )-2851875 x^{5}+11749725 x^{3}+15062970 x^{2}+7234552 x +1234232}{5400 x^{5}+17280 x^{4}+22104 x^{3}+14128 x^{2}+4512 x +576} \] Input:
int((1-2*x)/(2+3*x)^4/(3+5*x)^3,x)
Output:
(136890000*log(5*x + 3)*x**5 + 438048000*log(5*x + 3)*x**4 + 560336400*log (5*x + 3)*x**3 + 358144800*log(5*x + 3)*x**2 + 114379200*log(5*x + 3)*x + 14601600*log(5*x + 3) - 136890000*log(3*x + 2)*x**5 - 438048000*log(3*x + 2)*x**4 - 560336400*log(3*x + 2)*x**3 - 358144800*log(3*x + 2)*x**2 - 1143 79200*log(3*x + 2)*x - 14601600*log(3*x + 2) - 2851875*x**5 + 11749725*x** 3 + 15062970*x**2 + 7234552*x + 1234232)/(8*(675*x**5 + 2160*x**4 + 2763*x **3 + 1766*x**2 + 564*x + 72))