Integrand size = 22, antiderivative size = 66 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {4691 x}{15625}-\frac {7617 x^2}{6250}+\frac {2826 x^3}{3125}+\frac {513 x^4}{625}-\frac {648 x^5}{625}-\frac {1331}{781250 (3+5 x)^2}-\frac {15246}{390625 (3+5 x)}+\frac {63294 \log (3+5 x)}{390625} \] Output:
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/(1171875+1953125*x)+63294/390625*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {21586298+83293560 x+53587800 x^2-81707500 x^3+15815625 x^4+148050000 x^5-41343750 x^6-101250000 x^7+632940 (3+5 x)^2 \log (6 (3+5 x))}{3906250 (3+5 x)^2} \] Input:
Integrate[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^3,x]
Output:
(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 = 0.20 (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 |
\(\displaystyle -\frac {648 x^5}{625}+\frac {513 x^4}{625}+\frac {2826 x^3}{3125}-\frac {7617 x^2}{6250}+\frac {4691 x}{15625}-\frac {15246}{390625 (5 x+3)}-\frac {1331}{781250 (5 x+3)^2}+\frac {63294 \log (5 x+3)}{390625}\) |
Input:
Int[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^3,x]
Output:
(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
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 = 0.23 (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 | \(-\frac {648 x^{5}}{625}+\frac {513 x^{4}}{625}+\frac {2826 x^{3}}{3125}-\frac {7617 x^{2}}{6250}+\frac {4691 x}{15625}-\frac {15246}{390625 \left (3+5 x \right )}+\frac {63294 \ln \left (3+5 x \right )}{390625}-\frac {1331}{781250 \left (3+5 x \right )^{2}}\) | \(51\) |
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 | \(\frac {8 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {28 x \left (15 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {63294 \ln \left (1+\frac {5 x}{3}\right )}{390625}-\frac {14 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {1827 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{6250 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {567 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {729 x \left (-\frac {21875}{243} x^{5}+\frac {8750}{81} x^{4}-\frac {4375}{27} x^{3}+\frac {3500}{9} x^{2}+1050 x +420\right )}{15625 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {6561 x \left (\frac {125000}{729} x^{6}-\frac {43750}{243} x^{5}+\frac {17500}{81} x^{4}-\frac {8750}{27} x^{3}+\frac {7000}{9} x^{2}+2100 x +840\right )}{390625 \left (1+\frac {5 x}{3}\right )^{2}}\) | \(190\) |
Input:
int((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
-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 = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {20250000 \, x^{7} + 8268750 \, x^{6} - 29610000 \, x^{5} - 3163125 \, x^{4} + 16341500 \, x^{3} + 1532625 \, x^{2} - 126588 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 1958490 \, x + 92807}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:
integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")
Output:
-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 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=- \frac {648 x^{5}}{625} + \frac {513 x^{4}}{625} + \frac {2826 x^{3}}{3125} - \frac {7617 x^{2}}{6250} + \frac {4691 x}{15625} - \frac {152460 x + 92807}{19531250 x^{2} + 23437500 x + 7031250} + \frac {63294 \log {\left (5 x + 3 \right )}}{390625} \] Input:
integrate((1-2*x)**3*(2+3*x)**4/(3+5*x)**3,x)
Output:
-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 = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {648}{625} \, x^{5} + \frac {513}{625} \, x^{4} + \frac {2826}{3125} \, x^{3} - \frac {7617}{6250} \, x^{2} + \frac {4691}{15625} \, x - \frac {121 \, {\left (1260 \, x + 767\right )}}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {63294}{390625} \, \log \left (5 \, x + 3\right ) \] Input:
integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")
Output:
-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 = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {648}{625} \, x^{5} + \frac {513}{625} \, x^{4} + \frac {2826}{3125} \, x^{3} - \frac {7617}{6250} \, x^{2} + \frac {4691}{15625} \, x - \frac {121 \, {\left (1260 \, x + 767\right )}}{781250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {63294}{390625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \] Input:
integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")
Output:
-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 = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {4691\,x}{15625}+\frac {63294\,\ln \left (x+\frac {3}{5}\right )}{390625}-\frac {\frac {15246\,x}{1953125}+\frac {92807}{19531250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {7617\,x^2}{6250}+\frac {2826\,x^3}{3125}+\frac {513\,x^4}{625}-\frac {648\,x^5}{625} \] Input:
int(-((2*x - 1)^3*(3*x + 2)^4)/(5*x + 3)^3,x)
Output:
(4691*x)/15625 + (63294*log(x + 3/5))/390625 - ((15246*x)/1953125 + 92807/ 19531250)/((6*x)/5 + x^2 + 9/25) - (7617*x^2)/6250 + (2826*x^3)/3125 + (51 3*x^4)/625 - (648*x^5)/625
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.11 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {3164700 \,\mathrm {log}\left (5 x +3\right ) x^{2}+3797640 \,\mathrm {log}\left (5 x +3\right ) x +1139292 \,\mathrm {log}\left (5 x +3\right )-20250000 x^{7}-8268750 x^{6}+29610000 x^{5}+3163125 x^{4}-16341500 x^{3}-3164700 x^{2}-680354}{19531250 x^{2}+23437500 x +7031250} \] Input:
int((1-2*x)^3*(2+3*x)^4/(3+5*x)^3,x)
Output:
(3164700*log(5*x + 3)*x**2 + 3797640*log(5*x + 3)*x + 1139292*log(5*x + 3) - 20250000*x**7 - 8268750*x**6 + 29610000*x**5 + 3163125*x**4 - 16341500* x**3 - 3164700*x**2 - 680354)/(781250*(25*x**2 + 30*x + 9))