Integrand size = 22, antiderivative size = 87 \[ \int \frac {(1-2 x)^2 (2+3 x)^8}{(3+5 x)^3} \, dx=\frac {92582457 x}{9765625}+\frac {55559043 x^2}{3906250}-\frac {5350194 x^3}{390625}-\frac {1700919 x^4}{31250}-\frac {74601 x^5}{3125}+\frac {376407 x^6}{6250}+\frac {332424 x^7}{4375}+\frac {6561 x^8}{250}-\frac {121}{97656250 (3+5 x)^2}-\frac {572}{9765625 (3+5 x)}+\frac {5888 \log (3+5 x)}{9765625} \]
92582457/9765625*x+55559043/3906250*x^2-5350194/390625*x^3-1700919/31250*x ^4-74601/3125*x^5+376407/6250*x^6+332424/4375*x^7+6561/250*x^8-121/9765625 0/(3+5*x)^2-572/9765625/(3+5*x)+5888/9765625*ln(3+5*x)
Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^2 (2+3 x)^8}{(3+5 x)^3} \, dx=\frac {10358007077+92853841190 x+310701230325 x^2+369438720000 x^3-372682800000 x^4-1497169800000 x^5-1049233500000 x^6+1294582500000 x^7+2748937500000 x^8+1836738281250 x^9+448505859375 x^{10}+412160 (3+5 x)^2 \log (3+5 x)}{683593750 (3+5 x)^2} \]
(10358007077 + 92853841190*x + 310701230325*x^2 + 369438720000*x^3 - 37268 2800000*x^4 - 1497169800000*x^5 - 1049233500000*x^6 + 1294582500000*x^7 + 2748937500000*x^8 + 1836738281250*x^9 + 448505859375*x^10 + 412160*(3 + 5* x)^2*Log[3 + 5*x])/(683593750*(3 + 5*x)^2)
Time = 0.21 (sec) , antiderivative size = 87, 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.
\(\displaystyle \int \frac {(1-2 x)^2 (3 x+2)^8}{(5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {26244 x^7}{125}+\frac {332424 x^6}{625}+\frac {1129221 x^5}{3125}-\frac {74601 x^4}{625}-\frac {3401838 x^3}{15625}-\frac {16050582 x^2}{390625}+\frac {55559043 x}{1953125}+\frac {5888}{1953125 (5 x+3)}+\frac {572}{1953125 (5 x+3)^2}+\frac {121}{9765625 (5 x+3)^3}+\frac {92582457}{9765625}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6561 x^8}{250}+\frac {332424 x^7}{4375}+\frac {376407 x^6}{6250}-\frac {74601 x^5}{3125}-\frac {1700919 x^4}{31250}-\frac {5350194 x^3}{390625}+\frac {55559043 x^2}{3906250}+\frac {92582457 x}{9765625}-\frac {572}{9765625 (5 x+3)}-\frac {121}{97656250 (5 x+3)^2}+\frac {5888 \log (5 x+3)}{9765625}\) |
(92582457*x)/9765625 + (55559043*x^2)/3906250 - (5350194*x^3)/390625 - (17 00919*x^4)/31250 - (74601*x^5)/3125 + (376407*x^6)/6250 + (332424*x^7)/437 5 + (6561*x^8)/250 - 121/(97656250*(3 + 5*x)^2) - 572/(9765625*(3 + 5*x)) + (5888*Log[3 + 5*x])/9765625
3.14.21.3.1 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 = 2.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {6561 x^{8}}{250}+\frac {332424 x^{7}}{4375}+\frac {376407 x^{6}}{6250}-\frac {74601 x^{5}}{3125}-\frac {1700919 x^{4}}{31250}-\frac {5350194 x^{3}}{390625}+\frac {55559043 x^{2}}{3906250}+\frac {92582457 x}{9765625}+\frac {-\frac {572 x}{1953125}-\frac {17281}{97656250}}{\left (3+5 x \right )^{2}}+\frac {5888 \ln \left (3+5 x \right )}{9765625}\) | \(62\) |
default | \(\frac {92582457 x}{9765625}+\frac {55559043 x^{2}}{3906250}-\frac {5350194 x^{3}}{390625}-\frac {1700919 x^{4}}{31250}-\frac {74601 x^{5}}{3125}+\frac {376407 x^{6}}{6250}+\frac {332424 x^{7}}{4375}+\frac {6561 x^{8}}{250}-\frac {121}{97656250 \left (3+5 x \right )^{2}}-\frac {572}{9765625 \left (3+5 x \right )}+\frac {5888 \ln \left (3+5 x \right )}{9765625}\) | \(66\) |
norman | \(\frac {\frac {499947008}{5859375} x +\frac {1449920512}{3515625} x^{2}+\frac {42221568}{78125} x^{3}-\frac {8518464}{15625} x^{4}-\frac {34221024}{15625} x^{5}-\frac {4796496}{3125} x^{6}+\frac {8285328}{4375} x^{7}+\frac {703728}{175} x^{8}+\frac {94041}{35} x^{9}+\frac {6561}{10} x^{10}}{\left (3+5 x \right )^{2}}+\frac {5888 \ln \left (3+5 x \right )}{9765625}\) | \(67\) |
parallelrisch | \(\frac {807310546875 x^{10}+3306128906250 x^{9}+4948087500000 x^{8}+2330248500000 x^{7}-1888620300000 x^{6}-2694905640000 x^{5}-670829040000 x^{4}+18547200 \ln \left (x +\frac {3}{5}\right ) x^{2}+664989696000 x^{3}+22256640 \ln \left (x +\frac {3}{5}\right ) x +507472179200 x^{2}+6676992 \ln \left (x +\frac {3}{5}\right )+104988871680 x}{1230468750 \left (3+5 x \right )^{2}}\) | \(81\) |
meijerg | \(\frac {41452398 x \left (\frac {3906250}{6561} x^{8}-\frac {390625}{729} x^{7}+\frac {125000}{243} x^{6}-\frac {43750}{81} x^{5}+\frac {17500}{27} x^{4}-\frac {8750}{9} x^{3}+\frac {7000}{3} x^{2}+6300 x +2520\right )}{68359375 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {5888 \ln \left (1+\frac {5 x}{3}\right )}{9765625}-\frac {14348907 x \left (-\frac {150390625}{19683} x^{9}+\frac {42968750}{6561} x^{8}-\frac {4296875}{729} x^{7}+\frac {1375000}{243} x^{6}-\frac {481250}{81} x^{5}+\frac {192500}{27} x^{4}-\frac {96250}{9} x^{3}+\frac {77000}{3} x^{2}+69300 x +27720\right )}{1503906250 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {1043199 x \left (-\frac {390625}{729} x^{7}+\frac {125000}{243} x^{6}-\frac {43750}{81} x^{5}+\frac {17500}{27} x^{4}-\frac {8750}{9} x^{3}+\frac {7000}{3} x^{2}+6300 x +2520\right )}{781250 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {128 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {2432 x \left (15 x +6\right )}{225 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {8748 x \left (-\frac {21875}{243} x^{5}+\frac {8750}{81} x^{4}-\frac {4375}{27} x^{3}+\frac {3500}{9} x^{2}+1050 x +420\right )}{15625 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {1456542 x \left (\frac {125000}{729} x^{6}-\frac {43750}{243} x^{5}+\frac {17500}{81} x^{4}-\frac {8750}{27} x^{3}+\frac {7000}{9} x^{2}+2100 x +840\right )}{390625 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {192 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{25 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {1024 x^{2}}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {57456 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {18144 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{625 \left (1+\frac {5 x}{3}\right )^{2}}\) | \(352\) |
6561/250*x^8+332424/4375*x^7+376407/6250*x^6-74601/3125*x^5-1700919/31250* x^4-5350194/390625*x^3+55559043/3906250*x^2+92582457/9765625*x+25*(-572/48 828125*x-17281/2441406250)/(3+5*x)^2+5888/9765625*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^2 (2+3 x)^8}{(3+5 x)^3} \, dx=\frac {448505859375 \, x^{10} + 1836738281250 \, x^{9} + 2748937500000 \, x^{8} + 1294582500000 \, x^{7} - 1049233500000 \, x^{6} - 1497169800000 \, x^{5} - 372682800000 \, x^{4} + 369438720000 \, x^{3} + 281928652425 \, x^{2} + 412160 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 58326747710 \, x - 120967}{683593750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
1/683593750*(448505859375*x^10 + 1836738281250*x^9 + 2748937500000*x^8 + 1 294582500000*x^7 - 1049233500000*x^6 - 1497169800000*x^5 - 372682800000*x^ 4 + 369438720000*x^3 + 281928652425*x^2 + 412160*(25*x^2 + 30*x + 9)*log(5 *x + 3) + 58326747710*x - 120967)/(25*x^2 + 30*x + 9)
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^2 (2+3 x)^8}{(3+5 x)^3} \, dx=\frac {6561 x^{8}}{250} + \frac {332424 x^{7}}{4375} + \frac {376407 x^{6}}{6250} - \frac {74601 x^{5}}{3125} - \frac {1700919 x^{4}}{31250} - \frac {5350194 x^{3}}{390625} + \frac {55559043 x^{2}}{3906250} + \frac {92582457 x}{9765625} + \frac {- 28600 x - 17281}{2441406250 x^{2} + 2929687500 x + 878906250} + \frac {5888 \log {\left (5 x + 3 \right )}}{9765625} \]
6561*x**8/250 + 332424*x**7/4375 + 376407*x**6/6250 - 74601*x**5/3125 - 17 00919*x**4/31250 - 5350194*x**3/390625 + 55559043*x**2/3906250 + 92582457* x/9765625 + (-28600*x - 17281)/(2441406250*x**2 + 2929687500*x + 878906250 ) + 5888*log(5*x + 3)/9765625
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^2 (2+3 x)^8}{(3+5 x)^3} \, dx=\frac {6561}{250} \, x^{8} + \frac {332424}{4375} \, x^{7} + \frac {376407}{6250} \, x^{6} - \frac {74601}{3125} \, x^{5} - \frac {1700919}{31250} \, x^{4} - \frac {5350194}{390625} \, x^{3} + \frac {55559043}{3906250} \, x^{2} + \frac {92582457}{9765625} \, x - \frac {11 \, {\left (2600 \, x + 1571\right )}}{97656250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {5888}{9765625} \, \log \left (5 \, x + 3\right ) \]
6561/250*x^8 + 332424/4375*x^7 + 376407/6250*x^6 - 74601/3125*x^5 - 170091 9/31250*x^4 - 5350194/390625*x^3 + 55559043/3906250*x^2 + 92582457/9765625 *x - 11/97656250*(2600*x + 1571)/(25*x^2 + 30*x + 9) + 5888/9765625*log(5* x + 3)
Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^2 (2+3 x)^8}{(3+5 x)^3} \, dx=\frac {6561}{250} \, x^{8} + \frac {332424}{4375} \, x^{7} + \frac {376407}{6250} \, x^{6} - \frac {74601}{3125} \, x^{5} - \frac {1700919}{31250} \, x^{4} - \frac {5350194}{390625} \, x^{3} + \frac {55559043}{3906250} \, x^{2} + \frac {92582457}{9765625} \, x - \frac {11 \, {\left (2600 \, x + 1571\right )}}{97656250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {5888}{9765625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]
6561/250*x^8 + 332424/4375*x^7 + 376407/6250*x^6 - 74601/3125*x^5 - 170091 9/31250*x^4 - 5350194/390625*x^3 + 55559043/3906250*x^2 + 92582457/9765625 *x - 11/97656250*(2600*x + 1571)/(5*x + 3)^2 + 5888/9765625*log(abs(5*x + 3))
Time = 1.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^2 (2+3 x)^8}{(3+5 x)^3} \, dx=\frac {92582457\,x}{9765625}+\frac {5888\,\ln \left (x+\frac {3}{5}\right )}{9765625}-\frac {\frac {572\,x}{48828125}+\frac {17281}{2441406250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}+\frac {55559043\,x^2}{3906250}-\frac {5350194\,x^3}{390625}-\frac {1700919\,x^4}{31250}-\frac {74601\,x^5}{3125}+\frac {376407\,x^6}{6250}+\frac {332424\,x^7}{4375}+\frac {6561\,x^8}{250} \]