Integrand size = 22, antiderivative size = 76 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^2} \, dx=\frac {27776932 x}{1953125}+\frac {17592879 x^2}{781250}-\frac {1512378 x^3}{78125}-\frac {213867 x^4}{2500}-\frac {656424 x^5}{15625}+\frac {116397 x^6}{1250}+\frac {107892 x^7}{875}+\frac {2187 x^8}{50}-\frac {121}{9765625 (3+5 x)}+\frac {2497 \log (3+5 x)}{9765625} \]
27776932/1953125*x+17592879/781250*x^2-1512378/78125*x^3-213867/2500*x^4-6 56424/15625*x^5+116397/1250*x^6+107892/875*x^7+2187/50*x^8-121/9765625/(3+ 5*x)+2497/9765625*ln(3+5*x)
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^2} \, dx=\frac {9997654777+74994343395 x+189581876750 x^2+74537846250 x^3-483208621875 x^4-757103878125 x^5+94742156250 x^6+1142289843750 x^7+1022308593750 x^8+299003906250 x^9+349580 (3+5 x) \log (3+5 x)}{1367187500 (3+5 x)} \]
(9997654777 + 74994343395*x + 189581876750*x^2 + 74537846250*x^3 - 4832086 21875*x^4 - 757103878125*x^5 + 94742156250*x^6 + 1142289843750*x^7 + 10223 08593750*x^8 + 299003906250*x^9 + 349580*(3 + 5*x)*Log[3 + 5*x])/(13671875 00*(3 + 5*x))
Time = 0.20 (sec) , antiderivative size = 76, 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.
| \(\displaystyle \int \frac {(1-2 x)^2 (3 x+2)^7}{(5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {8748 x^7}{25}+\frac {107892 x^6}{125}+\frac {349191 x^5}{625}-\frac {656424 x^4}{3125}-\frac {213867 x^3}{625}-\frac {4537134 x^2}{78125}+\frac {17592879 x}{390625}+\frac {2497}{1953125 (5 x+3)}+\frac {121}{1953125 (5 x+3)^2}+\frac {27776932}{1953125}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2187 x^8}{50}+\frac {107892 x^7}{875}+\frac {116397 x^6}{1250}-\frac {656424 x^5}{15625}-\frac {213867 x^4}{2500}-\frac {1512378 x^3}{78125}+\frac {17592879 x^2}{781250}+\frac {27776932 x}{1953125}-\frac {121}{9765625 (5 x+3)}+\frac {2497 \log (5 x+3)}{9765625}\) |
(27776932*x)/1953125 + (17592879*x^2)/781250 - (1512378*x^3)/78125 - (2138 67*x^4)/2500 - (656424*x^5)/15625 + (116397*x^6)/1250 + (107892*x^7)/875 + (2187*x^8)/50 - 121/(9765625*(3 + 5*x)) + (2497*Log[3 + 5*x])/9765625
3.14.5.3.1 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 = 2.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {2187 x^{8}}{50}+\frac {107892 x^{7}}{875}+\frac {116397 x^{6}}{1250}-\frac {656424 x^{5}}{15625}-\frac {213867 x^{4}}{2500}-\frac {1512378 x^{3}}{78125}+\frac {17592879 x^{2}}{781250}+\frac {27776932 x}{1953125}-\frac {121}{48828125 \left (x +\frac {3}{5}\right )}+\frac {2497 \ln \left (3+5 x \right )}{9765625}\) | \(55\) |
| default | \(\frac {27776932 x}{1953125}+\frac {17592879 x^{2}}{781250}-\frac {1512378 x^{3}}{78125}-\frac {213867 x^{4}}{2500}-\frac {656424 x^{5}}{15625}+\frac {116397 x^{6}}{1250}+\frac {107892 x^{7}}{875}+\frac {2187 x^{8}}{50}-\frac {121}{9765625 \left (3+5 x \right )}+\frac {2497 \ln \left (3+5 x \right )}{9765625}\) | \(57\) |
| norman | \(\frac {\frac {249992509}{5859375} x +\frac {108332501}{781250} x^{2}+\frac {8518611}{156250} x^{3}-\frac {22089537}{62500} x^{4}-\frac {34610463}{62500} x^{5}+\frac {433107}{6250} x^{6}+\frac {1462131}{1750} x^{7}+\frac {261711}{350} x^{8}+\frac {2187}{10} x^{9}}{3+5 x}+\frac {2497 \ln \left (3+5 x \right )}{9765625}\) | \(62\) |
| parallelrisch | \(\frac {179402343750 x^{9}+613385156250 x^{8}+685373906250 x^{7}+56845293750 x^{6}-454262326875 x^{5}-289925173125 x^{4}+44722707750 x^{3}+1048740 \ln \left (x +\frac {3}{5}\right ) x +113749126050 x^{2}+629244 \ln \left (x +\frac {3}{5}\right )+34998951260 x}{2460937500+4101562500 x}\) | \(67\) |
| meijerg | \(-\frac {1856 x}{45 \left (1+\frac {5 x}{3}\right )}+\frac {2497 \ln \left (1+\frac {5 x}{3}\right )}{9765625}+\frac {1184 x \left (5 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )}+\frac {2772 x \left (-\frac {50}{9} x^{2}+10 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )}-\frac {40824 x \left (\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )}+\frac {22113 x \left (-\frac {625}{27} x^{4}+\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )}+\frac {111537 x \left (\frac {43750}{243} x^{5}-\frac {4375}{27} x^{4}+\frac {4375}{27} x^{3}-\frac {1750}{9} x^{2}+350 x +420\right )}{78125 \left (1+\frac {5 x}{3}\right )}-\frac {3483891 x \left (-\frac {312500}{729} x^{6}+\frac {87500}{243} x^{5}-\frac {8750}{27} x^{4}+\frac {8750}{27} x^{3}-\frac {3500}{9} x^{2}+700 x +840\right )}{3125000 \left (1+\frac {5 x}{3}\right )}+\frac {2598156 x \left (\frac {390625}{243} x^{7}-\frac {312500}{243} x^{6}+\frac {87500}{81} x^{5}-\frac {8750}{9} x^{4}+\frac {8750}{9} x^{3}-\frac {3500}{3} x^{2}+2100 x +2520\right )}{13671875 \left (1+\frac {5 x}{3}\right )}-\frac {4782969 x \left (-\frac {13671875}{6561} x^{8}+\frac {390625}{243} x^{7}-\frac {312500}{243} x^{6}+\frac {87500}{81} x^{5}-\frac {8750}{9} x^{4}+\frac {8750}{9} x^{3}-\frac {3500}{3} x^{2}+2100 x +2520\right )}{136718750 \left (1+\frac {5 x}{3}\right )}\) | \(280\) |
2187/50*x^8+107892/875*x^7+116397/1250*x^6-656424/15625*x^5-213867/2500*x^ 4-1512378/78125*x^3+17592879/781250*x^2+27776932/1953125*x-121/48828125/(x +3/5)+2497/9765625*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^2} \, dx=\frac {59800781250 \, x^{9} + 204461718750 \, x^{8} + 228457968750 \, x^{7} + 18948431250 \, x^{6} - 151420775625 \, x^{5} - 96641724375 \, x^{4} + 14907569250 \, x^{3} + 37916375350 \, x^{2} + 69916 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 11666311440 \, x - 3388}{273437500 \, {\left (5 \, x + 3\right )}} \]
1/273437500*(59800781250*x^9 + 204461718750*x^8 + 228457968750*x^7 + 18948 431250*x^6 - 151420775625*x^5 - 96641724375*x^4 + 14907569250*x^3 + 379163 75350*x^2 + 69916*(5*x + 3)*log(5*x + 3) + 11666311440*x - 3388)/(5*x + 3)
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^2} \, dx=\frac {2187 x^{8}}{50} + \frac {107892 x^{7}}{875} + \frac {116397 x^{6}}{1250} - \frac {656424 x^{5}}{15625} - \frac {213867 x^{4}}{2500} - \frac {1512378 x^{3}}{78125} + \frac {17592879 x^{2}}{781250} + \frac {27776932 x}{1953125} + \frac {2497 \log {\left (5 x + 3 \right )}}{9765625} - \frac {121}{48828125 x + 29296875} \]
2187*x**8/50 + 107892*x**7/875 + 116397*x**6/1250 - 656424*x**5/15625 - 21 3867*x**4/2500 - 1512378*x**3/78125 + 17592879*x**2/781250 + 27776932*x/19 53125 + 2497*log(5*x + 3)/9765625 - 121/(48828125*x + 29296875)
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^2} \, dx=\frac {2187}{50} \, x^{8} + \frac {107892}{875} \, x^{7} + \frac {116397}{1250} \, x^{6} - \frac {656424}{15625} \, x^{5} - \frac {213867}{2500} \, x^{4} - \frac {1512378}{78125} \, x^{3} + \frac {17592879}{781250} \, x^{2} + \frac {27776932}{1953125} \, x - \frac {121}{9765625 \, {\left (5 \, x + 3\right )}} + \frac {2497}{9765625} \, \log \left (5 \, x + 3\right ) \]
2187/50*x^8 + 107892/875*x^7 + 116397/1250*x^6 - 656424/15625*x^5 - 213867 /2500*x^4 - 1512378/78125*x^3 + 17592879/781250*x^2 + 27776932/1953125*x - 121/9765625/(5*x + 3) + 2497/9765625*log(5*x + 3)
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.34 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^2} \, dx=-\frac {1}{1367187500} \, {\left (5 \, x + 3\right )}^{8} {\left (\frac {1516320}{5 \, x + 3} - \frac {1411830}{{\left (5 \, x + 3\right )}^{2}} - \frac {11319588}{{\left (5 \, x + 3\right )}^{3}} - \frac {17377605}{{\left (5 \, x + 3\right )}^{4}} - \frac {14103180}{{\left (5 \, x + 3\right )}^{5}} - \frac {7427910}{{\left (5 \, x + 3\right )}^{6}} - \frac {3072860}{{\left (5 \, x + 3\right )}^{7}} - 153090\right )} - \frac {121}{9765625 \, {\left (5 \, x + 3\right )}} - \frac {2497}{9765625} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]
-1/1367187500*(5*x + 3)^8*(1516320/(5*x + 3) - 1411830/(5*x + 3)^2 - 11319 588/(5*x + 3)^3 - 17377605/(5*x + 3)^4 - 14103180/(5*x + 3)^5 - 7427910/(5 *x + 3)^6 - 3072860/(5*x + 3)^7 - 153090) - 121/9765625/(5*x + 3) - 2497/9 765625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^2} \, dx=\frac {27776932\,x}{1953125}+\frac {2497\,\ln \left (x+\frac {3}{5}\right )}{9765625}-\frac {121}{48828125\,\left (x+\frac {3}{5}\right )}+\frac {17592879\,x^2}{781250}-\frac {1512378\,x^3}{78125}-\frac {213867\,x^4}{2500}-\frac {656424\,x^5}{15625}+\frac {116397\,x^6}{1250}+\frac {107892\,x^7}{875}+\frac {2187\,x^8}{50} \]