Integrand size = 22, antiderivative size = 109 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx=\frac {21296}{5764801 (1-2 x)}+\frac {1}{3087 (2+3 x)^7}-\frac {101}{18522 (2+3 x)^6}+\frac {363}{12005 (2+3 x)^5}-\frac {3267}{67228 (2+3 x)^4}-\frac {4840}{117649 (2+3 x)^3}-\frac {22506}{823543 (2+3 x)^2}-\frac {17424}{823543 (2+3 x)}-\frac {307824 \log (1-2 x)}{40353607}+\frac {307824 \log (2+3 x)}{40353607} \]
21296/5764801/(1-2*x)+1/3087/(2+3*x)^7-101/18522/(2+3*x)^6+363/12005/(2+3* x)^5-3267/67228/(2+3*x)^4-4840/117649/(2+3*x)^3-22506/823543/(2+3*x)^2-174 24/823543/(2+3*x)-307824/40353607*ln(1-2*x)+307824/40353607*ln(2+3*x)
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.68 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx=\frac {4 \left (-\frac {7 \left (-8381276704-39853850134 x-18916696050 x^2+242725322763 x^3+677745912690 x^4+820756518120 x^5+494810149680 x^6+121177995840 x^7\right )}{16 (-1+2 x) (2+3 x)^7}-10389060 \log (1-2 x)+10389060 \log (4+6 x)\right )}{5447736945} \]
(4*((-7*(-8381276704 - 39853850134*x - 18916696050*x^2 + 242725322763*x^3 + 677745912690*x^4 + 820756518120*x^5 + 494810149680*x^6 + 121177995840*x^ 7))/(16*(-1 + 2*x)*(2 + 3*x)^7) - 10389060*Log[1 - 2*x] + 10389060*Log[4 + 6*x]))/5447736945
Time = 0.24 (sec) , antiderivative size = 109, 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.
| \(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^2 (3 x+2)^8} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {923472}{40353607 (3 x+2)}+\frac {52272}{823543 (3 x+2)^2}+\frac {135036}{823543 (3 x+2)^3}+\frac {43560}{117649 (3 x+2)^4}+\frac {9801}{16807 (3 x+2)^5}-\frac {1089}{2401 (3 x+2)^6}+\frac {101}{1029 (3 x+2)^7}-\frac {1}{147 (3 x+2)^8}-\frac {615648}{40353607 (2 x-1)}+\frac {42592}{5764801 (2 x-1)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {21296}{5764801 (1-2 x)}-\frac {17424}{823543 (3 x+2)}-\frac {22506}{823543 (3 x+2)^2}-\frac {4840}{117649 (3 x+2)^3}-\frac {3267}{67228 (3 x+2)^4}+\frac {363}{12005 (3 x+2)^5}-\frac {101}{18522 (3 x+2)^6}+\frac {1}{3087 (3 x+2)^7}-\frac {307824 \log (1-2 x)}{40353607}+\frac {307824 \log (3 x+2)}{40353607}\) |
21296/(5764801*(1 - 2*x)) + 1/(3087*(2 + 3*x)^7) - 101/(18522*(2 + 3*x)^6) + 363/(12005*(2 + 3*x)^5) - 3267/(67228*(2 + 3*x)^4) - 4840/(117649*(2 + 3*x)^3) - 22506/(823543*(2 + 3*x)^2) - 17424/(823543*(2 + 3*x)) - (307824* Log[1 - 2*x])/40353607 + (307824*Log[2 + 3*x])/40353607
3.16.87.3.1 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 = 2.69 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62
| method | result | size |
| norman | \(\frac {-\frac {1284260967}{16470860} x^{3}-\frac {358595721}{1647086} x^{4}-\frac {224403696}{5764801} x^{7}-\frac {217131354}{823543} x^{5}-\frac {18700308}{117649} x^{6}+\frac {90079505}{14823774} x^{2}+\frac {2846703581}{222356610} x +\frac {2095319176}{778248135}}{\left (-1+2 x \right ) \left (2+3 x \right )^{7}}-\frac {307824 \ln \left (-1+2 x \right )}{40353607}+\frac {307824 \ln \left (2+3 x \right )}{40353607}\) | \(68\) |
| risch | \(\frac {-\frac {1284260967}{16470860} x^{3}-\frac {358595721}{1647086} x^{4}-\frac {224403696}{5764801} x^{7}-\frac {217131354}{823543} x^{5}-\frac {18700308}{117649} x^{6}+\frac {90079505}{14823774} x^{2}+\frac {2846703581}{222356610} x +\frac {2095319176}{778248135}}{\left (-1+2 x \right ) \left (2+3 x \right )^{7}}-\frac {307824 \ln \left (-1+2 x \right )}{40353607}+\frac {307824 \ln \left (2+3 x \right )}{40353607}\) | \(69\) |
| default | \(-\frac {21296}{5764801 \left (-1+2 x \right )}-\frac {307824 \ln \left (-1+2 x \right )}{40353607}+\frac {1}{3087 \left (2+3 x \right )^{7}}-\frac {101}{18522 \left (2+3 x \right )^{6}}+\frac {363}{12005 \left (2+3 x \right )^{5}}-\frac {3267}{67228 \left (2+3 x \right )^{4}}-\frac {4840}{117649 \left (2+3 x \right )^{3}}-\frac {22506}{823543 \left (2+3 x \right )^{2}}-\frac {17424}{823543 \left (2+3 x \right )}+\frac {307824 \ln \left (2+3 x \right )}{40353607}\) | \(90\) |
| parallelrisch | \(\frac {-260395867200 x -595750256640 \ln \left (\frac {2}{3}+x \right ) x^{3}-661944729600 \ln \left (\frac {2}{3}+x \right ) x^{2}-214344007680 \ln \left (\frac {2}{3}+x \right ) x +6743285168928 x^{5}+12527552006928 x^{6}+8895054548520 x^{7}-3656451430240 x^{3}-1515955320320 x^{4}-1668317300160 x^{2}+2376091945584 x^{8}-1489375641600 \ln \left (x -\frac {1}{2}\right ) x^{4}+1489375641600 \ln \left (\frac {2}{3}+x \right ) x^{4}-25216942080 \ln \left (\frac {2}{3}+x \right )+595750256640 \ln \left (x -\frac {1}{2}\right ) x^{3}+3590459136000 \ln \left (\frac {2}{3}+x \right ) x^{7}+661944729600 \ln \left (x -\frac {1}{2}\right ) x^{2}+214344007680 \ln \left (x -\frac {1}{2}\right ) x +4914939617280 \ln \left (\frac {2}{3}+x \right ) x^{5}+6031971348480 \ln \left (\frac {2}{3}+x \right ) x^{6}-861710192640 \ln \left (x -\frac {1}{2}\right ) x^{8}+25216942080 \ln \left (x -\frac {1}{2}\right )+861710192640 \ln \left (\frac {2}{3}+x \right ) x^{8}-3590459136000 \ln \left (x -\frac {1}{2}\right ) x^{7}-6031971348480 \ln \left (x -\frac {1}{2}\right ) x^{6}-4914939617280 \ln \left (x -\frac {1}{2}\right ) x^{5}}{25826308480 \left (-1+2 x \right ) \left (2+3 x \right )^{7}}\) | \(208\) |
(-1284260967/16470860*x^3-358595721/1647086*x^4-224403696/5764801*x^7-2171 31354/823543*x^5-18700308/117649*x^6+90079505/14823774*x^2+2846703581/2223 56610*x+2095319176/778248135)/(-1+2*x)/(2+3*x)^7-307824/40353607*ln(-1+2*x )+307824/40353607*ln(2+3*x)
Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.61 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx=-\frac {848245970880 \, x^{7} + 3463671047760 \, x^{6} + 5745295626840 \, x^{5} + 4744221388830 \, x^{4} + 1699077259341 \, x^{3} - 132416872350 \, x^{2} - 166224960 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )} \log \left (3 \, x + 2\right ) + 166224960 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )} \log \left (2 \, x - 1\right ) - 278976950938 \, x - 58668936928}{21790947780 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )}} \]
-1/21790947780*(848245970880*x^7 + 3463671047760*x^6 + 5745295626840*x^5 + 4744221388830*x^4 + 1699077259341*x^3 - 132416872350*x^2 - 166224960*(437 4*x^8 + 18225*x^7 + 30618*x^6 + 24948*x^5 + 7560*x^4 - 3024*x^3 - 3360*x^2 - 1088*x - 128)*log(3*x + 2) + 166224960*(4374*x^8 + 18225*x^7 + 30618*x^ 6 + 24948*x^5 + 7560*x^4 - 3024*x^3 - 3360*x^2 - 1088*x - 128)*log(2*x - 1 ) - 278976950938*x - 58668936928)/(4374*x^8 + 18225*x^7 + 30618*x^6 + 2494 8*x^5 + 7560*x^4 - 3024*x^3 - 3360*x^2 - 1088*x - 128)
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx=\frac {- 121177995840 x^{7} - 494810149680 x^{6} - 820756518120 x^{5} - 677745912690 x^{4} - 242725322763 x^{3} + 18916696050 x^{2} + 39853850134 x + 8381276704}{13616229369960 x^{8} + 56734289041500 x^{7} + 95313605589720 x^{6} + 77662937887920 x^{5} + 23534223602400 x^{4} - 9413689440960 x^{3} - 10459654934400 x^{2} - 3386935883520 x - 398463045120} - \frac {307824 \log {\left (x - \frac {1}{2} \right )}}{40353607} + \frac {307824 \log {\left (x + \frac {2}{3} \right )}}{40353607} \]
(-121177995840*x**7 - 494810149680*x**6 - 820756518120*x**5 - 677745912690 *x**4 - 242725322763*x**3 + 18916696050*x**2 + 39853850134*x + 8381276704) /(13616229369960*x**8 + 56734289041500*x**7 + 95313605589720*x**6 + 776629 37887920*x**5 + 23534223602400*x**4 - 9413689440960*x**3 - 10459654934400* x**2 - 3386935883520*x - 398463045120) - 307824*log(x - 1/2)/40353607 + 30 7824*log(x + 2/3)/40353607
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx=-\frac {121177995840 \, x^{7} + 494810149680 \, x^{6} + 820756518120 \, x^{5} + 677745912690 \, x^{4} + 242725322763 \, x^{3} - 18916696050 \, x^{2} - 39853850134 \, x - 8381276704}{3112992540 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )}} + \frac {307824}{40353607} \, \log \left (3 \, x + 2\right ) - \frac {307824}{40353607} \, \log \left (2 \, x - 1\right ) \]
-1/3112992540*(121177995840*x^7 + 494810149680*x^6 + 820756518120*x^5 + 67 7745912690*x^4 + 242725322763*x^3 - 18916696050*x^2 - 39853850134*x - 8381 276704)/(4374*x^8 + 18225*x^7 + 30618*x^6 + 24948*x^5 + 7560*x^4 - 3024*x^ 3 - 3360*x^2 - 1088*x - 128) + 307824/40353607*log(3*x + 2) - 307824/40353 607*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx=-\frac {21296}{5764801 \, {\left (2 \, x - 1\right )}} + \frac {4 \, {\left (\frac {108987508287}{2 \, x - 1} + \frac {677288963799}{{\left (2 \, x - 1\right )}^{2}} + \frac {2255033089785}{{\left (2 \, x - 1\right )}^{3}} + \frac {4241269979800}{{\left (2 \, x - 1\right )}^{4}} + \frac {4269658683500}{{\left (2 \, x - 1\right )}^{5}} + \frac {1795850807520}{{\left (2 \, x - 1\right )}^{6}} + 7339564629\right )}}{1412376245 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{7}} + \frac {307824}{40353607} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]
-21296/5764801/(2*x - 1) + 4/1412376245*(108987508287/(2*x - 1) + 67728896 3799/(2*x - 1)^2 + 2255033089785/(2*x - 1)^3 + 4241269979800/(2*x - 1)^4 + 4269658683500/(2*x - 1)^5 + 1795850807520/(2*x - 1)^6 + 7339564629)/(7/(2 *x - 1) + 3)^7 + 307824/40353607*log(abs(-7/(2*x - 1) - 3))
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.79 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx=\frac {615648\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{40353607}-\frac {\frac {51304\,x^7}{5764801}+\frac {12826\,x^6}{352947}+\frac {1340317\,x^5}{22235661}+\frac {13281323\,x^4}{266827932}+\frac {47565221\,x^3}{2668279320}-\frac {90079505\,x^2}{64839187476}-\frac {2846703581\,x}{972587812140}-\frac {1047659588}{1702028671245}}{x^8+\frac {25\,x^7}{6}+7\,x^6+\frac {154\,x^5}{27}+\frac {140\,x^4}{81}-\frac {56\,x^3}{81}-\frac {560\,x^2}{729}-\frac {544\,x}{2187}-\frac {64}{2187}} \]
(615648*atanh((12*x)/7 + 1/7))/40353607 - ((47565221*x^3)/2668279320 - (90 079505*x^2)/64839187476 - (2846703581*x)/972587812140 + (13281323*x^4)/266 827932 + (1340317*x^5)/22235661 + (12826*x^6)/352947 + (51304*x^7)/5764801 - 1047659588/1702028671245)/((140*x^4)/81 - (560*x^2)/729 - (56*x^3)/81 - (544*x)/2187 + (154*x^5)/27 + 7*x^6 + (25*x^7)/6 + x^8 - 64/2187)