3.17.97 \(\int \frac {(2+3 x)^9}{(1-2 x)^3 (3+5 x)^3} \, dx\) [1697]

3.17.97.1 Optimal result

Integrand size = 22, antiderivative size = 91 \[ \int \frac {(2+3 x)^9}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {40353607}{340736 (1-2 x)^2}-\frac {17294403}{29282 (1-2 x)}-\frac {50150097 x}{100000}-\frac {7459857 x^2}{50000}-\frac {373977 x^3}{10000}-\frac {19683 x^4}{4000}-\frac {1}{207968750 (3+5 x)^2}-\frac {303}{1143828125 (3+5 x)}-\frac {12657032367 \log (1-2 x)}{20614528}+\frac {8202 \log (3+5 x)}{2516421875} \]

output

40353607/340736/(1-2*x)^2-17294403/29282/(1-2*x)-50150097/100000*x-7459857 
/50000*x^2-373977/10000*x^3-19683/4000*x^4-1/207968750/(3+5*x)^2-303/11438 
28125/(3+5*x)-12657032367/20614528*ln(1-2*x)+8202/2516421875*ln(3+5*x)
 

3.17.97.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82 \[ \int \frac {(2+3 x)^9}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {-\frac {11 \left (1977372328510687+1867950230356442 x-12213049363361937 x^2-15846035365304040 x^3+8450955823285800 x^4+14903967293820000 x^5+4502793796875000 x^6+1123897331700000 x^7+144089401500000 x^8\right )}{\left (-3+x+10 x^2\right )^2}-1977661307343750 \log (3-6 x)[phone] \log (-3 (3+5 x))}{3221020000000} \]

input

Integrate[(2 + 3*x)^9/((1 - 2*x)^3*(3 + 5*x)^3),x]
 

output

((-11*(1977372328510687 + 1867950230356442*x - 12213049363361937*x^2 - 158 
46035365304040*x^3 + 8450955823285800*x^4 + 14903967293820000*x^5 + 450279 
3796875000*x^6 + 1123897331700000*x^7 + 144089401500000*x^8))/(-3 + x + 10 
*x^2)^2 - 1977661307343750*Log[3 - 6*x] + 10498560*Log[-3*(3 + 5*x)])/3221 
020000000
 

3.17.97.3 Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 91, 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.

\(\displaystyle \int \frac {(3 x+2)^9}{(1-2 x)^3 (5 x+3)^3} \, dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {19683 x^3}{1000}-\frac {1121931 x^2}{10000}-\frac {7459857 x}{25000}-\frac {12657032367}{10307264 (2 x-1)}+\frac {8202}{503284375 (5 x+3)}-\frac {17294403}{14641 (2 x-1)^2}+\frac {303}{228765625 (5 x+3)^2}-\frac {40353607}{85184 (2 x-1)^3}+\frac {1}{20796875 (5 x+3)^3}-\frac {50150097}{100000}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {19683 x^4}{4000}-\frac {373977 x^3}{10000}-\frac {7459857 x^2}{50000}-\frac {50150097 x}{100000}-\frac {17294403}{29282 (1-2 x)}-\frac {303}{1143828125 (5 x+3)}+\frac {40353607}{340736 (1-2 x)^2}-\frac {1}{207968750 (5 x+3)^2}-\frac {12657032367 \log (1-2 x)}{20614528}+\frac {8202 \log (5 x+3)}{2516421875}\)

input

Int[(2 + 3*x)^9/((1 - 2*x)^3*(3 + 5*x)^3),x]
 

output

40353607/(340736*(1 - 2*x)^2) - 17294403/(29282*(1 - 2*x)) - (50150097*x)/ 
100000 - (7459857*x^2)/50000 - (373977*x^3)/10000 - (19683*x^4)/4000 - 1/( 
207968750*(3 + 5*x)^2) - 303/(1143828125*(3 + 5*x)) - (12657032367*Log[1 - 
 2*x])/20614528 + (8202*Log[3 + 5*x])/2516421875
 

3.17.97.3.1 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]
 

3.17.97.4 Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {19683 x^{4}}{4000}-\frac {373977 x^{3}}{10000}-\frac {7459857 x^{2}}{50000}-\frac {50150097 x}{100000}+\frac {\frac {6755626170663}{228765625} x^{3}+\frac {6920013076005537}{292820000000} x^{2}-\frac {517480964491321}{146410000000} x -\frac {1244386341093487}{292820000000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {12657032367 \ln \left (-1+2 x \right )}{20614528}+\frac {8202 \ln \left (3+5 x \right )}{2516421875}\)	\(67\)
norman	\(\frac {-\frac {237500757975359}{29282000000} x +\frac {43840566324903}{732050000} x^{3}+\frac {1445397085524663}{58564000000} x^{2}-\frac {50898051}{1000} x^{5}-\frac {492075}{32} x^{6}-\frac {767637}{200} x^{7}-\frac {19683}{40} x^{8}-\frac {243357260882993}{58564000000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {12657032367 \ln \left (-1+2 x \right )}{20614528}+\frac {8202 \ln \left (3+5 x \right )}{2516421875}\)	\(68\)
default	\(-\frac {19683 x^{4}}{4000}-\frac {373977 x^{3}}{10000}-\frac {7459857 x^{2}}{50000}-\frac {50150097 x}{100000}-\frac {1}{207968750 \left (3+5 x \right )^{2}}-\frac {303}{1143828125 \left (3+5 x \right )}+\frac {8202 \ln \left (3+5 x \right )}{2516421875}+\frac {40353607}{340736 \left (-1+2 x \right )^{2}}+\frac {17294403}{29282 \left (-1+2 x \right )}-\frac {12657032367 \ln \left (-1+2 x \right )}{20614528}\)	\(72\)
parallelrisch	\(\frac {-475495024950000 x^{8}-3708861194610000 x^{7}-14859219529687500 x^{6}[phone] \ln \left (x +\frac {3}{5}\right ) x^{4}-59329839220312500 \ln \left (x -\frac {1}{2}\right ) x^{4}-49183092069606000 x^{5}-15771682324349655+62991360 \ln \left (x +\frac {3}{5}\right ) x^{3}-11865967844062500 \ln \left (x -\frac {1}{2}\right ) x^{3}-130625416886447450 x^{4}-185824512 \ln \left (x +\frac {3}{5}\right ) x^{2}+35004605139984375 \ln \left (x -\frac {1}{2}\right ) x^{2}+31744464171582470 x^{3}-18897408 \ln \left (x +\frac {3}{5}\right ) x +3559790353218750 \ln \left (x -\frac {1}{2}\right ) x +100918047874160935 x^{2}[phone] \ln \left (x +\frac {3}{5}\right )-5339685529828125 \ln \left (x -\frac {1}{2}\right )}{966306000000 \left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}\)	\(134\)

input

int((2+3*x)^9/(1-2*x)^3/(3+5*x)^3,x,method=_RETURNVERBOSE)
 

output

-19683/4000*x^4-373977/10000*x^3-7459857/50000*x^2-50150097/100000*x+100*( 
6755626170663/22876562500*x^3+6920013076005537/29282000000000*x^2-51748096 
4491321/14641000000000*x-1244386341093487/29282000000000)/(-1+2*x)^2/(3+5* 
x)^2-12657032367/20614528*ln(-1+2*x)+8202/2516421875*ln(3+5*x)
 

3.17.97.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.32 \[ \int \frac {(2+3 x)^9}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {1584983416500000 \, x^{8} + 12362870648700000 \, x^{7} + 49530731765625000 \, x^{6} + 163943640232020000 \, x^{5} + 3373337816263800 \, x^{4} - 192223824266320440 \, x^{3} - 81487109015452107 \, x^{2} - 10498560 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 1977661307343750 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 25922683108313662 \, x + 13688249752028357}{3221020000000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

input

integrate((2+3*x)^9/(1-2*x)^3/(3+5*x)^3,x, algorithm="fricas")
 

output

-1/3221020000000*(1584983416500000*x^8 + 12362870648700000*x^7 + 495307317 
65625000*x^6 + 163943640232020000*x^5 + 3373337816263800*x^4 - 19222382426 
6320440*x^3 - 81487109015452107*x^2 - 10498560*(100*x^4 + 20*x^3 - 59*x^2 
- 6*x + 9)*log(5*x + 3) + 1977661307343750*(100*x^4 + 20*x^3 - 59*x^2 - 6* 
x + 9)*log(2*x - 1) + 25922683108313662*x + 13688249752028357)/(100*x^4 + 
20*x^3 - 59*x^2 - 6*x + 9)
 

3.17.97.6 Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^9}{(1-2 x)^3 (3+5 x)^3} \, dx=- \frac {19683 x^{4}}{4000} - \frac {373977 x^{3}}{10000} - \frac {7459857 x^{2}}{50000} - \frac {50150097 x}{100000} - \frac {- 8647201498448640 x^{3} - 6920013076005537 x^{2} + 1034961928982642 x + 1244386341093487}{29282000000000 x^{4} + 5856400000000 x^{3} - 17276380000000 x^{2} - 1756920000000 x + 2635380000000} - \frac {12657032367 \log {\left (x - \frac {1}{2} \right )}}{20614528} + \frac {8202 \log {\left (x + \frac {3}{5} \right )}}{2516421875} \]

input

integrate((2+3*x)**9/(1-2*x)**3/(3+5*x)**3,x)
 

output

-19683*x**4/4000 - 373977*x**3/10000 - 7459857*x**2/50000 - 50150097*x/100 
000 - (-8647201498448640*x**3 - 6920013076005537*x**2 + 1034961928982642*x 
 + 1244386341093487)/(29282000000000*x**4 + 5856400000000*x**3 - 172763800 
00000*x**2 - 1756920000000*x + 2635380000000) - 12657032367*log(x - 1/2)/2 
0614528 + 8202*log(x + 3/5)/2516421875
 

3.17.97.7 Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.81 \[ \int \frac {(2+3 x)^9}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {19683}{4000} \, x^{4} - \frac {373977}{10000} \, x^{3} - \frac {7459857}{50000} \, x^{2} - \frac {50150097}{100000} \, x + \frac {8647201498448640 \, x^{3} + 6920013076005537 \, x^{2} - 1034961928982642 \, x - 1244386341093487}{292820000000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {8202}{2516421875} \, \log \left (5 \, x + 3\right ) - \frac {12657032367}{20614528} \, \log \left (2 \, x - 1\right ) \]

input

integrate((2+3*x)^9/(1-2*x)^3/(3+5*x)^3,x, algorithm="maxima")
 

output

-19683/4000*x^4 - 373977/10000*x^3 - 7459857/50000*x^2 - 50150097/100000*x 
 + 1/292820000000*(8647201498448640*x^3 + 6920013076005537*x^2 - 103496192 
8982642*x - 1244386341093487)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 8202 
/2516421875*log(5*x + 3) - 12657032367/20614528*log(2*x - 1)
 

3.17.97.8 Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^9}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {19683}{4000} \, x^{4} - \frac {373977}{10000} \, x^{3} - \frac {7459857}{50000} \, x^{2} - \frac {50150097}{100000} \, x + \frac {8647201498448640 \, x^{3} + 6920013076005537 \, x^{2} - 1034961928982642 \, x - 1244386341093487}{292820000000 \, {\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {8202}{2516421875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {12657032367}{20614528} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

input

integrate((2+3*x)^9/(1-2*x)^3/(3+5*x)^3,x, algorithm="giac")
 

output

-19683/4000*x^4 - 373977/10000*x^3 - 7459857/50000*x^2 - 50150097/100000*x 
 + 1/292820000000*(8647201498448640*x^3 + 6920013076005537*x^2 - 103496192 
8982642*x - 1244386341093487)/((5*x + 3)^2*(2*x - 1)^2) + 8202/2516421875* 
log(abs(5*x + 3)) - 12657032367/20614528*log(abs(2*x - 1))
 

3.17.97.9 Mupad [B] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^9}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {8202\,\ln \left (x+\frac {3}{5}\right )}{2516421875}-\frac {12657032367\,\ln \left (x-\frac {1}{2}\right )}{20614528}-\frac {50150097\,x}{100000}-\frac {-\frac {6755626170663\,x^3}{22876562500}-\frac {6920013076005537\,x^2}{29282000000000}+\frac {517480964491321\,x}{14641000000000}+\frac {1244386341093487}{29282000000000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}}-\frac {7459857\,x^2}{50000}-\frac {373977\,x^3}{10000}-\frac {19683\,x^4}{4000} \]

input

int(-(3*x + 2)^9/((2*x - 1)^3*(5*x + 3)^3),x)
 

output

(8202*log(x + 3/5))/2516421875 - (12657032367*log(x - 1/2))/20614528 - (50 
150097*x)/100000 - ((517480964491321*x)/14641000000000 - (6920013076005537 
*x^2)/29282000000000 - (6755626170663*x^3)/22876562500 + 1244386341093487/ 
29282000000000)/(x^3/5 - (59*x^2)/100 - (3*x)/50 + x^4 + 9/100) - (7459857 
*x^2)/50000 - (373977*x^3)/10000 - (19683*x^4)/4000