Integrand size = 22, antiderivative size = 62 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {133659 x}{78125}-\frac {1816 x^2}{625}-\frac {4217 x^3}{3125}+\frac {7317 x^4}{1250}+\frac {108 x^5}{625}-\frac {108 x^6}{25}-\frac {1331}{390625 (3+5 x)}+\frac {15246 \log (3+5 x)}{390625} \] Output:
133659/78125*x-1816/625*x^2-4217/3125*x^3+7317/1250*x^4+108/625*x^5-108/25 *x^6-1331/(1171875+1953125*x)+15246/390625*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {14487499+44216865 x-635250 x^2-72563750 x^3+42240625 x^4+116353125 x^5-47250000 x^6-84375000 x^7+152460 (3+5 x) \log (6 (3+5 x))}{3906250 (3+5 x)} \] Input:
Integrate[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^2,x]
Output:
(14487499 + 44216865*x - 635250*x^2 - 72563750*x^3 + 42240625*x^4 + 116353 125*x^5 - 47250000*x^6 - 84375000*x^7 + 152460*(3 + 5*x)*Log[6*(3 + 5*x)]) /(3906250*(3 + 5*x))
Time = 0.23 (sec) , antiderivative size = 62, 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.
\(\displaystyle \int \frac {(1-2 x)^3 (3 x+2)^4}{(5 x+3)^2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {648 x^5}{25}+\frac {108 x^4}{125}+\frac {14634 x^3}{625}-\frac {12651 x^2}{3125}-\frac {3632 x}{625}+\frac {15246}{78125 (5 x+3)}+\frac {1331}{78125 (5 x+3)^2}+\frac {133659}{78125}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {108 x^6}{25}+\frac {108 x^5}{625}+\frac {7317 x^4}{1250}-\frac {4217 x^3}{3125}-\frac {1816 x^2}{625}+\frac {133659 x}{78125}-\frac {1331}{390625 (5 x+3)}+\frac {15246 \log (5 x+3)}{390625}\) |
Input:
Int[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^2,x]
Output:
(133659*x)/78125 - (1816*x^2)/625 - (4217*x^3)/3125 + (7317*x^4)/1250 + (1 08*x^5)/625 - (108*x^6)/25 - 1331/(390625*(3 + 5*x)) + (15246*Log[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 = 45, normalized size of antiderivative = 0.73
method | result | size |
risch | \(-\frac {108 x^{6}}{25}+\frac {108 x^{5}}{625}+\frac {7317 x^{4}}{1250}-\frac {4217 x^{3}}{3125}-\frac {1816 x^{2}}{625}+\frac {133659 x}{78125}-\frac {1331}{1953125 \left (x +\frac {3}{5}\right )}+\frac {15246 \ln \left (3+5 x \right )}{390625}\) | \(45\) |
default | \(-\frac {108 x^{6}}{25}+\frac {108 x^{5}}{625}+\frac {7317 x^{4}}{1250}-\frac {4217 x^{3}}{3125}-\frac {1816 x^{2}}{625}+\frac {133659 x}{78125}-\frac {1331}{390625 \left (3+5 x \right )}+\frac {15246 \ln \left (3+5 x \right )}{390625}\) | \(47\) |
norman | \(\frac {\frac {1204262}{234375} x -\frac {2541}{15625} x^{2}-\frac {58051}{3125} x^{3}+\frac {13517}{1250} x^{4}+\frac {37233}{1250} x^{5}-\frac {1512}{125} x^{6}-\frac {108}{5} x^{7}}{3+5 x}+\frac {15246 \ln \left (3+5 x \right )}{390625}\) | \(52\) |
parallelrisch | \(\frac {-50625000 x^{7}-28350000 x^{6}+69811875 x^{5}+25344375 x^{4}-43538250 x^{3}+457380 \ln \left (x +\frac {3}{5}\right ) x -381150 x^{2}+274428 \ln \left (x +\frac {3}{5}\right )+12042620 x}{7031250+11718750 x}\) | \(57\) |
meijerg | \(\frac {16 x}{9 \left (1+\frac {5 x}{3}\right )}-\frac {56 x \left (5 x +6\right )}{25 \left (1+\frac {5 x}{3}\right )}+\frac {15246 \ln \left (1+\frac {5 x}{3}\right )}{390625}+\frac {42 x \left (-\frac {50}{9} x^{2}+10 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )}+\frac {1827 x \left (\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )}-\frac {1701 x \left (-\frac {625}{27} x^{4}+\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{6250 \left (1+\frac {5 x}{3}\right )}-\frac {4374 x \left (\frac {43750}{243} x^{5}-\frac {4375}{27} x^{4}+\frac {4375}{27} x^{3}-\frac {1750}{9} x^{2}+350 x +420\right )}{78125 \left (1+\frac {5 x}{3}\right )}+\frac {6561 x \left (-\frac {312500}{729} x^{6}+\frac {87500}{243} x^{5}-\frac {8750}{27} x^{4}+\frac {8750}{27} x^{3}-\frac {3500}{9} x^{2}+700 x +840\right )}{390625 \left (1+\frac {5 x}{3}\right )}\) | \(185\) |
Input:
int((1-2*x)^3*(2+3*x)^4/(3+5*x)^2,x,method=_RETURNVERBOSE)
Output:
-108/25*x^6+108/625*x^5+7317/1250*x^4-4217/3125*x^3-1816/625*x^2+133659/78 125*x-1331/1953125/(x+3/5)+15246/390625*ln(3+5*x)
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {16875000 \, x^{7} + 9450000 \, x^{6} - 23270625 \, x^{5} - 8448125 \, x^{4} + 14512750 \, x^{3} + 127050 \, x^{2} - 30492 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 4009770 \, x + 2662}{781250 \, {\left (5 \, x + 3\right )}} \] Input:
integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^2,x, algorithm="fricas")
Output:
-1/781250*(16875000*x^7 + 9450000*x^6 - 23270625*x^5 - 8448125*x^4 + 14512 750*x^3 + 127050*x^2 - 30492*(5*x + 3)*log(5*x + 3) - 4009770*x + 2662)/(5 *x + 3)
Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=- \frac {108 x^{6}}{25} + \frac {108 x^{5}}{625} + \frac {7317 x^{4}}{1250} - \frac {4217 x^{3}}{3125} - \frac {1816 x^{2}}{625} + \frac {133659 x}{78125} + \frac {15246 \log {\left (5 x + 3 \right )}}{390625} - \frac {1331}{1953125 x + 1171875} \] Input:
integrate((1-2*x)**3*(2+3*x)**4/(3+5*x)**2,x)
Output:
-108*x**6/25 + 108*x**5/625 + 7317*x**4/1250 - 4217*x**3/3125 - 1816*x**2/ 625 + 133659*x/78125 + 15246*log(5*x + 3)/390625 - 1331/(1953125*x + 11718 75)
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {108}{25} \, x^{6} + \frac {108}{625} \, x^{5} + \frac {7317}{1250} \, x^{4} - \frac {4217}{3125} \, x^{3} - \frac {1816}{625} \, x^{2} + \frac {133659}{78125} \, x - \frac {1331}{390625 \, {\left (5 \, x + 3\right )}} + \frac {15246}{390625} \, \log \left (5 \, x + 3\right ) \] Input:
integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^2,x, algorithm="maxima")
Output:
-108/25*x^6 + 108/625*x^5 + 7317/1250*x^4 - 4217/3125*x^3 - 1816/625*x^2 + 133659/78125*x - 1331/390625/(5*x + 3) + 15246/390625*log(5*x + 3)
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.35 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {1}{3906250} \, {\left (5 \, x + 3\right )}^{6} {\left (\frac {19656}{5 \, x + 3} - \frac {112455}{{\left (5 \, x + 3\right )}^{2}} + \frac {121450}{{\left (5 \, x + 3\right )}^{3}} + \frac {530600}{{\left (5 \, x + 3\right )}^{4}} + \frac {632940}{{\left (5 \, x + 3\right )}^{5}} - 1080\right )} - \frac {1331}{390625 \, {\left (5 \, x + 3\right )}} - \frac {15246}{390625} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \] Input:
integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^2,x, algorithm="giac")
Output:
1/3906250*(5*x + 3)^6*(19656/(5*x + 3) - 112455/(5*x + 3)^2 + 121450/(5*x + 3)^3 + 530600/(5*x + 3)^4 + 632940/(5*x + 3)^5 - 1080) - 1331/390625/(5* x + 3) - 15246/390625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {133659\,x}{78125}+\frac {15246\,\ln \left (x+\frac {3}{5}\right )}{390625}-\frac {1331}{1953125\,\left (x+\frac {3}{5}\right )}-\frac {1816\,x^2}{625}-\frac {4217\,x^3}{3125}+\frac {7317\,x^4}{1250}+\frac {108\,x^5}{625}-\frac {108\,x^6}{25} \] Input:
int(-((2*x - 1)^3*(3*x + 2)^4)/(5*x + 3)^2,x)
Output:
(133659*x)/78125 + (15246*log(x + 3/5))/390625 - 1331/(1953125*(x + 3/5)) - (1816*x^2)/625 - (4217*x^3)/3125 + (7317*x^4)/1250 + (108*x^5)/625 - (10 8*x^6)/25
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {457380 \,\mathrm {log}\left (5 x +3\right ) x +274428 \,\mathrm {log}\left (5 x +3\right )-50625000 x^{7}-28350000 x^{6}+69811875 x^{5}+25344375 x^{4}-43538250 x^{3}-381150 x^{2}+12042620 x}{11718750 x +7031250} \] Input:
int((1-2*x)^3*(2+3*x)^4/(3+5*x)^2,x)
Output:
(457380*log(5*x + 3)*x + 274428*log(5*x + 3) - 50625000*x**7 - 28350000*x* *6 + 69811875*x**5 + 25344375*x**4 - 43538250*x**3 - 381150*x**2 + 1204262 0*x)/(2343750*(5*x + 3))