3.17.17 \(\int \frac {(2+3 x)^8}{(1-2 x)^2 (3+5 x)^3} \, dx\) [1617]

3.17.17.1 Optimal result

Integrand size = 22, antiderivative size = 80 \[ \int \frac {(2+3 x)^8}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {5764801}{85184 (1-2 x)}+\frac {7680987 x}{50000}+\frac {6093711 x^2}{100000}+\frac {12393 x^3}{625}+\frac {6561 x^4}{2000}-\frac {1}{18906250 (3+5 x)^2}-\frac {268}{103984375 (3+5 x)}+\frac {130943337 \log (1-2 x)}{937024}+\frac {6312 \log (3+5 x)}{228765625} \]

output

5764801/85184/(1-2*x)+7680987/50000*x+6093711/100000*x^2+12393/625*x^3+656 
1/2000*x^4-1/18906250/(3+5*x)^2-268/103984375/(3+5*x)[phone]/937024*ln( 
1-2*x)+6312/228765625*ln(3+5*x)
 

3.17.17.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92 \[ \int \frac {(2+3 x)^8}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {3 \left (\frac {11}{3} \left (536108166000+\frac {450375078125}{1-2 x}[phone] x+405536467050 x^2+131960664000 x^3+21831727500 x^4-\frac {352}{(3+5 x)^2}-\frac {17152}{3+5 x}\right )[phone] \log (3-6 x)+673280 \log (-3 (3+5 x))\right )}{73205000000} \]

input

Integrate[(2 + 3*x)^8/((1 - 2*x)^2*(3 + 5*x)^3),x]
 

output

(3*((11*(536108166000 + 450375078125/(1 - 2*x) + 1022339369700*x + 4055364 
67050*x^2 + 131960664000*x^3 + 21831727500*x^4 - 352/(3 + 5*x)^2 - 17152/( 
3 + 5*x)))/3 + 3409982734375*Log[3 - 6*x] + 673280*Log[-3*(3 + 5*x)]))/732 
05000000
 

3.17.17.3 Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 80, 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.

\(\displaystyle \int \frac {(3 x+2)^8}{(1-2 x)^2 (5 x+3)^3} \, dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {6561 x^3}{500}+\frac {37179 x^2}{625}+\frac {6093711 x}{50000}+\frac {130943337}{468512 (2 x-1)}+\frac {6312}{45753125 (5 x+3)}+\frac {5764801}{42592 (2 x-1)^2}+\frac {268}{20796875 (5 x+3)^2}+\frac {1}{1890625 (5 x+3)^3}+\frac {7680987}{50000}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6561 x^4}{2000}+\frac {12393 x^3}{625}+\frac {6093711 x^2}{100000}+\frac {7680987 x}{50000}+\frac {5764801}{85184 (1-2 x)}-\frac {268}{103984375 (5 x+3)}-\frac {1}{18906250 (5 x+3)^2}+\frac {130943337 \log (1-2 x)}{937024}+\frac {6312 \log (5 x+3)}{228765625}\)

input

Int[(2 + 3*x)^8/((1 - 2*x)^2*(3 + 5*x)^3),x]
 

output

5764801/(85184*(1 - 2*x)) + (7680987*x)/50000 + (6093711*x^2)/100000 + (12 
393*x^3)/625 + (6561*x^4)/2000 - 1/(18906250*(3 + 5*x)^2) - 268/(103984375 
*(3 + 5*x)) + (130943337*Log[1 - 2*x])/937024 + (6312*Log[3 + 5*x])/228765 
625
 

3.17.17.3.1 Defintions of rubi rules used

rule 99

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]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.17.17.4 Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {6561 x^{4}}{2000}+\frac {12393 x^{3}}{625}+\frac {6093711 x^{2}}{100000}+\frac {7680987 x}{50000}+\frac {-\frac {2251875424929}{1331000000} x^{2}-\frac {6755626180803}{3327500000} x -\frac {4053375651317}{6655000000}}{\left (-1+2 x \right ) \left (3+5 x \right )^{2}}+\frac {130943337 \ln \left (-1+2 x \right )}{937024}+\frac {6312 \ln \left (3+5 x \right )}{228765625}\)	\(62\)
default	\(\frac {6561 x^{4}}{2000}+\frac {12393 x^{3}}{625}+\frac {6093711 x^{2}}{100000}+\frac {7680987 x}{50000}-\frac {1}{18906250 \left (3+5 x \right )^{2}}-\frac {268}{103984375 \left (3+5 x \right )}+\frac {6312 \ln \left (3+5 x \right )}{228765625}-\frac {5764801}{85184 \left (-1+2 x \right )}+\frac {130943337 \ln \left (-1+2 x \right )}{937024}\)	\(63\)
norman	\(\frac {-\frac {7729272953531}{1197900000} x^{2}+\frac {259525466657}{239580000} x^{3}-\frac {1038468349229}{399300000} x +\frac {38185263}{4000} x^{4}+\frac {1480599}{400} x^{5}+\frac {442503}{400} x^{6}+\frac {6561}{40} x^{7}}{\left (-1+2 x \right ) \left (3+5 x \right )^{2}}+\frac {130943337 \ln \left (-1+2 x \right )}{937024}+\frac {6312 \ln \left (3+5 x \right )}{228765625}\)	\(67\)
parallelrisch	\(\frac {21613410225000 x^{7}[phone] x^{6}[phone] x^{5}[phone] \ln \left (x +\frac {3}{5}\right ) x^{3}[phone] \ln \left (x -\frac {1}{2}\right ) x^{3}+1257908480061750 x^{4}[phone] \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (x -\frac {1}{2}\right ) x^{2}[phone] x^{3}-43628544 \ln \left (x +\frac {3}{5}\right ) x -220966881187500 \ln \left (x -\frac {1}{2}\right ) x -850220024888410 x^{2}-32721408 \ln \left (x +\frac {3}{5}\right )-165725160890625 \ln \left (x -\frac {1}{2}\right )-342694555245570 x}{131769000000 \left (-1+2 x \right ) \left (3+5 x \right )^{2}}\)	\(113\)

input

int((2+3*x)^8/(1-2*x)^2/(3+5*x)^3,x,method=_RETURNVERBOSE)
 

output

6561/2000*x^4+12393/625*x^3+6093711/100000*x^2+7680987/50000*x+50*(-225187 
5424929/66550000000*x^2-6755626180803/166375000000*x-4053375651317/3327500 
00000)/(-1+2*x)/(3+5*x)^2+130943337/937024*ln(-1+2*x)+6312/228765625*ln(3+ 
5*x)
 

3.17.17.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25 \[ \int \frac {(2+3 x)^8}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {12007450125000 \, x^{7} + 80983580287500 \, x^{6} + 270968124487500 \, x^{5} + 698838044478750 \, x^{4} + 327005737947900 \, x^{3} - 298950055409445 \, x^{2} + 2019840 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 10229948203125 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) - 249835373577966 \, x - 44587132164487}{73205000000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

input

integrate((2+3*x)^8/(1-2*x)^2/(3+5*x)^3,x, algorithm="fricas")
 

output

1/73205000000*(12007450125000*x^7 + 80983580287500*x^6 + 270968124487500*x 
^5 + 698838044478750*x^4 + 327005737947900*x^3 - 298950055409445*x^2 + 201 
9840*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) + 10229948203125*(50*x^3 + 
35*x^2 - 12*x - 9)*log(2*x - 1) - 249835373577966*x - 44587132164487)/(50* 
x^3 + 35*x^2 - 12*x - 9)
 

3.17.17.6 Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^8}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {6561 x^{4}}{2000} + \frac {12393 x^{3}}{625} + \frac {6093711 x^{2}}{100000} + \frac {7680987 x}{50000} + \frac {- 11259377124645 x^{2} - 13511252361606 x - 4053375651317}{332750000000 x^{3} + 232925000000 x^{2} - 79860000000 x - 59895000000} + \frac {130943337 \log {\left (x - \frac {1}{2} \right )}}{937024} + \frac {6312 \log {\left (x + \frac {3}{5} \right )}}{228765625} \]

input

integrate((2+3*x)**8/(1-2*x)**2/(3+5*x)**3,x)
 

output

6561*x**4/2000 + 12393*x**3/625 + 6093711*x**2/100000 + 7680987*x/50000 + 
(-11259377124645*x**2 - 13511252361606*x - 4053375651317)/(332750000000*x* 
*3 + 232925000000*x**2 - 79860000000*x - 59895000000) + 130943337*log(x - 
1/2)/937024 + 6312*log(x + 3/5)/228765625
 

3.17.17.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {(2+3 x)^8}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {6561}{2000} \, x^{4} + \frac {12393}{625} \, x^{3} + \frac {6093711}{100000} \, x^{2} + \frac {7680987}{50000} \, x - \frac {11259377124645 \, x^{2} + 13511252361606 \, x + 4053375651317}{6655000000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac {6312}{228765625} \, \log \left (5 \, x + 3\right ) + \frac {130943337}{937024} \, \log \left (2 \, x - 1\right ) \]

input

integrate((2+3*x)^8/(1-2*x)^2/(3+5*x)^3,x, algorithm="maxima")
 

output

6561/2000*x^4 + 12393/625*x^3 + 6093711/100000*x^2 + 7680987/50000*x - 1/6 
655000000*(11259377124645*x^2 + 13511252361606*x + 4053375651317)/(50*x^3 
+ 35*x^2 - 12*x - 9) + 6312/228765625*log(5*x + 3) + 130943337/937024*log( 
2*x - 1)
 

3.17.17.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.40 \[ \int \frac {(2+3 x)^8}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {{\left (2 \, x - 1\right )}^{4} {\left (\frac {1230096557250}{2 \, x - 1} + \frac {11539159570125}{{\left (2 \, x - 1\right )}^{2}} + \frac {69299175042900}{{\left (2 \, x - 1\right )}^{3}} + \frac {182728002843460}{{\left (2 \, x - 1\right )}^{4}} + \frac {163740919200408}{{\left (2 \, x - 1\right )}^{5}} + 60037250625\right )}}{11712800000 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2}} - \frac {5764801}{85184 \, {\left (2 \, x - 1\right )}} - \frac {139743873}{1000000} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {6312}{228765625} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \]

input

integrate((2+3*x)^8/(1-2*x)^2/(3+5*x)^3,x, algorithm="giac")
 

output

1/11712800000*(2*x - 1)^4*(1230096557250/(2*x - 1) + 11539159570125/(2*x - 
 1)^2 + 69299175042900/(2*x - 1)^3 + 182728002843460/(2*x - 1)^4 + 1637409 
19200408/(2*x - 1)^5 + 60037250625)/(11/(2*x - 1) + 5)^2 - 5764801/85184/( 
2*x - 1) - 139743873/1000000*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 6312/2287 
65625*log(abs(-11/(2*x - 1) - 5))
 

3.17.17.9 Mupad [B] (verification not implemented)

Time = 1.43 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^8}{(1-2 x)^2 (3+5 x)^3} \, dx=\frac {7680987\,x}{50000}+\frac {130943337\,\ln \left (x-\frac {1}{2}\right )}{937024}+\frac {6312\,\ln \left (x+\frac {3}{5}\right )}{228765625}+\frac {\frac {2251875424929\,x^2}{66550000000}+\frac {6755626180803\,x}{166375000000}+\frac {4053375651317}{332750000000}}{-x^3-\frac {7\,x^2}{10}+\frac {6\,x}{25}+\frac {9}{50}}+\frac {6093711\,x^2}{100000}+\frac {12393\,x^3}{625}+\frac {6561\,x^4}{2000} \]

input

int((3*x + 2)^8/((2*x - 1)^2*(5*x + 3)^3),x)
 

output

(7680987*x)/50000 + (130943337*log(x - 1/2))/937024 + (6312*log(x + 3/5))/ 
228765625 + ((6755626180803*x)/166375000000 + (2251875424929*x^2)/66550000 
000 + 4053375651317/332750000000)/((6*x)/25 - (7*x^2)/10 - x^3 + 9/50) + ( 
6093711*x^2)/100000 + (12393*x^3)/625 + (6561*x^4)/2000