Integrand size = 22, antiderivative size = 73 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {823543}{15488 (1-2 x)^2}-\frac {18941489}{85184 (1-2 x)}-\frac {1258983 x}{10000}-\frac {108621 x^2}{4000}-\frac {729 x^3}{200}-\frac {1}{4159375 (3+5 x)}-\frac {87177909 \log (1-2 x)}{468512}+\frac {237 \log (3+5 x)}{45753125} \] Output:
823543/15488/(1-2*x)^2-18941489/(85184-170368*x)-1258983/10000*x-108621/40 00*x^2-729/200*x^3-1/(12478125+20796875*x)-87177909/468512*ln(1-2*x)+237/4 5753125*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {-\frac {22 \left (19763981131+25343933346 x-163837494156 x^2-172378468845 x^3+151415158950 x^4+34203039750 x^5+4851495000 x^6\right )}{(1-2 x)^2 (3+5 x)}-272430965625 \log (1-2 x)+7584 \log (6+10 x)}{1464100000} \] Input:
Integrate[(2 + 3*x)^7/((1 - 2*x)^3*(3 + 5*x)^2),x]
Output:
((-22*(19763981131 + 25343933346*x - 163837494156*x^2 - 172378468845*x^3 + 151415158950*x^4 + 34203039750*x^5 + 4851495000*x^6))/((1 - 2*x)^2*(3 + 5 *x)) - 272430965625*Log[1 - 2*x] + 7584*Log[6 + 10*x])/1464100000
Time = 0.23 (sec) , antiderivative size = 73, 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.
| \(\displaystyle \int \frac {(3 x+2)^7}{(1-2 x)^3 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {2187 x^2}{200}-\frac {108621 x}{2000}-\frac {87177909}{234256 (2 x-1)}+\frac {237}{9150625 (5 x+3)}-\frac {18941489}{42592 (2 x-1)^2}+\frac {1}{831875 (5 x+3)^2}-\frac {823543}{3872 (2 x-1)^3}-\frac {1258983}{10000}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {729 x^3}{200}-\frac {108621 x^2}{4000}-\frac {1258983 x}{10000}-\frac {18941489}{85184 (1-2 x)}-\frac {1}{4159375 (5 x+3)}+\frac {823543}{15488 (1-2 x)^2}-\frac {87177909 \log (1-2 x)}{468512}+\frac {237 \log (5 x+3)}{45753125}\) |
Input:
Int[(2 + 3*x)^7/((1 - 2*x)^3*(3 + 5*x)^2),x]
Output:
823543/(15488*(1 - 2*x)^2) - 18941489/(85184*(1 - 2*x)) - (1258983*x)/1000 0 - (108621*x^2)/4000 - (729*x^3)/200 - 1/(4159375*(3 + 5*x)) - (87177909* Log[1 - 2*x])/468512 + (237*Log[3 + 5*x])/45753125
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 = 57, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(-\frac {729 x^{3}}{200}-\frac {108621 x^{2}}{4000}-\frac {1258983 x}{10000}+\frac {\frac {295960765497}{133100000} x^{2}+\frac {259930759887}{532400000} x -\frac {270225047003}{532400000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {87177909 \ln \left (-1+2 x \right )}{468512}+\frac {237 \ln \left (3+5 x \right )}{45753125}\) | \(57\) |
| default | \(-\frac {729 x^{3}}{200}-\frac {108621 x^{2}}{4000}-\frac {1258983 x}{10000}-\frac {1}{4159375 \left (3+5 x \right )}+\frac {237 \ln \left (3+5 x \right )}{45753125}+\frac {823543}{15488 \left (-1+2 x \right )^{2}}+\frac {18941489}{85184 \left (-1+2 x \right )}-\frac {87177909 \ln \left (-1+2 x \right )}{468512}\) | \(58\) |
| norman | \(\frac {\frac {16670031671}{9982500} x^{2}-\frac {42875933591}{39930000} x +\frac {182483005831}{39930000} x^{3}-\frac {2275209}{1000} x^{4}-\frac {102789}{200} x^{5}-\frac {729}{10} x^{6}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {87177909 \ln \left (-1+2 x \right )}{468512}+\frac {237 \ln \left (3+5 x \right )}{45753125}\) | \(62\) |
| parallelrisch | \(-\frac {320198670000 x^{6}+2257400623500 x^{5}+16345857937500 \ln \left (x -\frac {1}{2}\right ) x^{3}-455040 \ln \left (x +\frac {3}{5}\right ) x^{3}+9993400490700 x^{4}-6538343175000 \ln \left (x -\frac {1}{2}\right ) x^{2}+182016 \ln \left (x +\frac {3}{5}\right ) x^{2}-20073130641410 x^{3}-5721050278125 \ln \left (x -\frac {1}{2}\right ) x +159264 \ln \left (x +\frac {3}{5}\right ) x -7334813935240 x^{2}+2451878690625 \ln \left (x -\frac {1}{2}\right )-68256 \ln \left (x +\frac {3}{5}\right )+4716352695010 x}{4392300000 \left (-1+2 x \right )^{2} \left (3+5 x \right )}\) | \(108\) |
Input:
int((2+3*x)^7/(1-2*x)^3/(3+5*x)^2,x,method=_RETURNVERBOSE)
Output:
-729/200*x^3-108621/4000*x^2-1258983/10000*x+20*(295960765497/2662000000*x ^2+259930759887/10648000000*x-270225047003/10648000000)/(-1+2*x)^2/(3+5*x) -87177909/468512*ln(-1+2*x)+237/45753125*ln(3+5*x)
Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.30 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {426931560000 \, x^{6} + 3009867498000 \, x^{5} + 13324533987600 \, x^{4} - 6947670741660 \, x^{3} - 17706353292408 \, x^{2} - 30336 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 1089723862500 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 647305946397 \, x + 2972475517033}{5856400000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \] Input:
integrate((2+3*x)^7/(1-2*x)^3/(3+5*x)^2,x, algorithm="fricas")
Output:
-1/5856400000*(426931560000*x^6 + 3009867498000*x^5 + 13324533987600*x^4 - 6947670741660*x^3 - 17706353292408*x^2 - 30336*(20*x^3 - 8*x^2 - 7*x + 3) *log(5*x + 3) + 1089723862500*(20*x^3 - 8*x^2 - 7*x + 3)*log(2*x - 1) - 64 7305946397*x + 2972475517033)/(20*x^3 - 8*x^2 - 7*x + 3)
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^2} \, dx=- \frac {729 x^{3}}{200} - \frac {108621 x^{2}}{4000} - \frac {1258983 x}{10000} - \frac {- 1183843061988 x^{2} - 259930759887 x + 270225047003}{10648000000 x^{3} - 4259200000 x^{2} - 3726800000 x + 1597200000} - \frac {87177909 \log {\left (x - \frac {1}{2} \right )}}{468512} + \frac {237 \log {\left (x + \frac {3}{5} \right )}}{45753125} \] Input:
integrate((2+3*x)**7/(1-2*x)**3/(3+5*x)**2,x)
Output:
-729*x**3/200 - 108621*x**2/4000 - 1258983*x/10000 - (-1183843061988*x**2 - 259930759887*x + 270225047003)/(10648000000*x**3 - 4259200000*x**2 - 372 6800000*x + 1597200000) - 87177909*log(x - 1/2)/468512 + 237*log(x + 3/5)/ 45753125
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {729}{200} \, x^{3} - \frac {108621}{4000} \, x^{2} - \frac {1258983}{10000} \, x + \frac {1183843061988 \, x^{2} + 259930759887 \, x - 270225047003}{532400000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac {237}{45753125} \, \log \left (5 \, x + 3\right ) - \frac {87177909}{468512} \, \log \left (2 \, x - 1\right ) \] Input:
integrate((2+3*x)^7/(1-2*x)^3/(3+5*x)^2,x, algorithm="maxima")
Output:
-729/200*x^3 - 108621/4000*x^2 - 1258983/10000*x + 1/532400000*(1183843061 988*x^2 + 259930759887*x - 270225047003)/(20*x^3 - 8*x^2 - 7*x + 3) + 237/ 45753125*log(5*x + 3) - 87177909/468512*log(2*x - 1)
Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {{\left (5 \, x + 3\right )}^{3} {\left (\frac {1472913882}{5 \, x + 3} + \frac {33001809588}{{\left (5 \, x + 3\right )}^{2}} - \frac {817302548083}{{\left (5 \, x + 3\right )}^{3}} + \frac {2996736348771}{{\left (5 \, x + 3\right )}^{4}} + 85386312\right )}}{732050000 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2}} - \frac {1}{4159375 \, {\left (5 \, x + 3\right )}} + \frac {18607401}{100000} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {87177909}{468512} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \] Input:
integrate((2+3*x)^7/(1-2*x)^3/(3+5*x)^2,x, algorithm="giac")
Output:
-1/732050000*(5*x + 3)^3*(1472913882/(5*x + 3) + 33001809588/(5*x + 3)^2 - 817302548083/(5*x + 3)^3 + 2996736348771/(5*x + 3)^4 + 85386312)/(11/(5*x + 3) - 2)^2 - 1/4159375/(5*x + 3) + 18607401/100000*log(1/5*abs(5*x + 3)/ (5*x + 3)^2) - 87177909/468512*log(abs(-11/(5*x + 3) + 2))
Time = 1.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {237\,\ln \left (x+\frac {3}{5}\right )}{45753125}-\frac {87177909\,\ln \left (x-\frac {1}{2}\right )}{468512}-\frac {1258983\,x}{10000}-\frac {\frac {295960765497\,x^2}{2662000000}+\frac {259930759887\,x}{10648000000}-\frac {270225047003}{10648000000}}{-x^3+\frac {2\,x^2}{5}+\frac {7\,x}{20}-\frac {3}{20}}-\frac {108621\,x^2}{4000}-\frac {729\,x^3}{200} \] Input:
int(-(3*x + 2)^7/((2*x - 1)^3*(5*x + 3)^2),x)
Output:
(237*log(x + 3/5))/45753125 - (87177909*log(x - 1/2))/468512 - (1258983*x) /10000 - ((259930759887*x)/10648000000 + (295960765497*x^2)/2662000000 - 2 70225047003/10648000000)/((7*x)/20 + (2*x^2)/5 - x^3 - 3/20) - (108621*x^2 )/4000 - (729*x^3)/200
Time = 0.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.66 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {151680 \,\mathrm {log}\left (5 x +3\right ) x^{3}-60672 \,\mathrm {log}\left (5 x +3\right ) x^{2}-53088 \,\mathrm {log}\left (5 x +3\right ) x +22752 \,\mathrm {log}\left (5 x +3\right )-5448619312500 \,\mathrm {log}\left (2 x -1\right ) x^{3}+2179447725000 \,\mathrm {log}\left (2 x -1\right ) x^{2}+1907016759375 \,\mathrm {log}\left (2 x -1\right ) x -817292896875 \,\mathrm {log}\left (2 x -1\right )-106732890000 x^{6}-752466874500 x^{5}-3331133496900 x^{4}+12803388493170 x^{3}-3711438296115 x +916851741905}{29282000000 x^{3}-11712800000 x^{2}-10248700000 x +4392300000} \] Input:
int((2+3*x)^7/(1-2*x)^3/(3+5*x)^2,x)
Output:
(151680*log(5*x + 3)*x**3 - 60672*log(5*x + 3)*x**2 - 53088*log(5*x + 3)*x + 22752*log(5*x + 3) - 5448619312500*log(2*x - 1)*x**3 + 2179447725000*lo g(2*x - 1)*x**2 + 1907016759375*log(2*x - 1)*x - 817292896875*log(2*x - 1) - 106732890000*x**6 - 752466874500*x**5 - 3331133496900*x**4 + 1280338849 3170*x**3 - 3711438296115*x + 916851741905)/(1464100000*(20*x**3 - 8*x**2 - 7*x + 3))