Integrand size = 22, antiderivative size = 66 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {4691 x}{15625}-\frac {7617 x^2}{6250}+\frac {2826 x^3}{3125}+\frac {513 x^4}{625}-\frac {648 x^5}{625}-\frac {1331}{781250 (3+5 x)^2}-\frac {15246}{390625 (3+5 x)}+\frac {63294 \log (3+5 x)}{390625} \]
4691/15625*x-7617/6250*x^2+2826/3125*x^3+513/625*x^4-648/625*x^5-1331/7812 50/(3+5*x)^2-15246/390625/(3+5*x)+63294/390625*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {21586298+83293560 x+53587800 x^2-81707500 x^3+15815625 x^4+148050000 x^5-41343750 x^6-101250000 x^7+632940 (3+5 x)^2 \log (6 (3+5 x))}{3906250 (3+5 x)^2} \]
(21586298 + 83293560*x + 53587800*x^2 - 81707500*x^3 + 15815625*x^4 + 1480 50000*x^5 - 41343750*x^6 - 101250000*x^7 + 632940*(3 + 5*x)^2*Log[6*(3 + 5 *x)])/(3906250*(3 + 5*x)^2)
Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int \frac {(1-2 x)^3 (3 x+2)^4}{(5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {648 x^4}{125}+\frac {2052 x^3}{625}+\frac {8478 x^2}{3125}-\frac {7617 x}{3125}+\frac {63294}{78125 (5 x+3)}+\frac {15246}{78125 (5 x+3)^2}+\frac {1331}{78125 (5 x+3)^3}+\frac {4691}{15625}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {648 x^5}{625}+\frac {513 x^4}{625}+\frac {2826 x^3}{3125}-\frac {7617 x^2}{6250}+\frac {4691 x}{15625}-\frac {15246}{390625 (5 x+3)}-\frac {1331}{781250 (5 x+3)^2}+\frac {63294 \log (5 x+3)}{390625}\) |
(4691*x)/15625 - (7617*x^2)/6250 + (2826*x^3)/3125 + (513*x^4)/625 - (648* x^5)/625 - 1331/(781250*(3 + 5*x)^2) - 15246/(390625*(3 + 5*x)) + (63294*L og[3 + 5*x])/390625
3.15.21.3.1 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 = 0.81 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.71
method | result | size |
risch | \(-\frac {648 x^{5}}{625}+\frac {513 x^{4}}{625}+\frac {2826 x^{3}}{3125}-\frac {7617 x^{2}}{6250}+\frac {4691 x}{15625}+\frac {-\frac {15246 x}{78125}-\frac {92807}{781250}}{\left (3+5 x \right )^{2}}+\frac {63294 \ln \left (3+5 x \right )}{390625}\) | \(47\) |
default | \(\frac {4691 x}{15625}-\frac {7617 x^{2}}{6250}+\frac {2826 x^{3}}{3125}+\frac {513 x^{4}}{625}-\frac {648 x^{5}}{625}-\frac {1331}{781250 \left (3+5 x \right )^{2}}-\frac {15246}{390625 \left (3+5 x \right )}+\frac {63294 \ln \left (3+5 x \right )}{390625}\) | \(51\) |
norman | \(\frac {\frac {680354}{234375} x -\frac {229469}{140625} x^{2}-\frac {65366}{3125} x^{3}+\frac {5061}{1250} x^{4}+\frac {23688}{625} x^{5}-\frac {1323}{125} x^{6}-\frac {648}{25} x^{7}}{\left (3+5 x \right )^{2}}+\frac {63294 \ln \left (3+5 x \right )}{390625}\) | \(52\) |
parallelrisch | \(\frac {-182250000 x^{7}-74418750 x^{6}+266490000 x^{5}+28468125 x^{4}+28482300 \ln \left (x +\frac {3}{5}\right ) x^{2}-147073500 x^{3}+34178760 \ln \left (x +\frac {3}{5}\right ) x -11473450 x^{2}+10253628 \ln \left (x +\frac {3}{5}\right )+20410620 x}{7031250 \left (3+5 x \right )^{2}}\) | \(66\) |
meijerg | \(\frac {8 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {28 x \left (15 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {63294 \ln \left (1+\frac {5 x}{3}\right )}{390625}-\frac {14 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {1827 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{6250 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {567 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {729 x \left (-\frac {21875}{243} x^{5}+\frac {8750}{81} x^{4}-\frac {4375}{27} x^{3}+\frac {3500}{9} x^{2}+1050 x +420\right )}{15625 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {6561 x \left (\frac {125000}{729} x^{6}-\frac {43750}{243} x^{5}+\frac {17500}{81} x^{4}-\frac {8750}{27} x^{3}+\frac {7000}{9} x^{2}+2100 x +840\right )}{390625 \left (1+\frac {5 x}{3}\right )^{2}}\) | \(190\) |
-648/625*x^5+513/625*x^4+2826/3125*x^3-7617/6250*x^2+4691/15625*x+25*(-152 46/1953125*x-92807/19531250)/(3+5*x)^2+63294/390625*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {20250000 \, x^{7} + 8268750 \, x^{6} - 29610000 \, x^{5} - 3163125 \, x^{4} + 16341500 \, x^{3} + 1532625 \, x^{2} - 126588 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 1958490 \, x + 92807}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
-1/781250*(20250000*x^7 + 8268750*x^6 - 29610000*x^5 - 3163125*x^4 + 16341 500*x^3 + 1532625*x^2 - 126588*(25*x^2 + 30*x + 9)*log(5*x + 3) - 1958490* x + 92807)/(25*x^2 + 30*x + 9)
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=- \frac {648 x^{5}}{625} + \frac {513 x^{4}}{625} + \frac {2826 x^{3}}{3125} - \frac {7617 x^{2}}{6250} + \frac {4691 x}{15625} - \frac {152460 x + 92807}{19531250 x^{2} + 23437500 x + 7031250} + \frac {63294 \log {\left (5 x + 3 \right )}}{390625} \]
-648*x**5/625 + 513*x**4/625 + 2826*x**3/3125 - 7617*x**2/6250 + 4691*x/15 625 - (152460*x + 92807)/(19531250*x**2 + 23437500*x + 7031250) + 63294*lo g(5*x + 3)/390625
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {648}{625} \, x^{5} + \frac {513}{625} \, x^{4} + \frac {2826}{3125} \, x^{3} - \frac {7617}{6250} \, x^{2} + \frac {4691}{15625} \, x - \frac {121 \, {\left (1260 \, x + 767\right )}}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {63294}{390625} \, \log \left (5 \, x + 3\right ) \]
-648/625*x^5 + 513/625*x^4 + 2826/3125*x^3 - 7617/6250*x^2 + 4691/15625*x - 121/781250*(1260*x + 767)/(25*x^2 + 30*x + 9) + 63294/390625*log(5*x + 3 )
Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {648}{625} \, x^{5} + \frac {513}{625} \, x^{4} + \frac {2826}{3125} \, x^{3} - \frac {7617}{6250} \, x^{2} + \frac {4691}{15625} \, x - \frac {121 \, {\left (1260 \, x + 767\right )}}{781250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {63294}{390625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]
-648/625*x^5 + 513/625*x^4 + 2826/3125*x^3 - 7617/6250*x^2 + 4691/15625*x - 121/781250*(1260*x + 767)/(5*x + 3)^2 + 63294/390625*log(abs(5*x + 3))
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {4691\,x}{15625}+\frac {63294\,\ln \left (x+\frac {3}{5}\right )}{390625}-\frac {\frac {15246\,x}{1953125}+\frac {92807}{19531250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {7617\,x^2}{6250}+\frac {2826\,x^3}{3125}+\frac {513\,x^4}{625}-\frac {648\,x^5}{625} \]