3.16.86 \(\int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^7} \, dx\) [1586]

3.16.86.1 Optimal result

Integrand size = 22, antiderivative size = 98 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^7} \, dx=\frac {10648}{823543 (1-2 x)}+\frac {1}{2646 (2+3 x)^6}-\frac {101}{15435 (2+3 x)^5}+\frac {363}{9604 (2+3 x)^4}-\frac {1089}{16807 (2+3 x)^3}-\frac {7260}{117649 (2+3 x)^2}-\frac {45012}{823543 (2+3 x)}-\frac {17424 \log (1-2 x)}{823543}+\frac {17424 \log (2+3 x)}{823543} \]

output

10648/823543/(1-2*x)+1/2646/(2+3*x)^6-101/15435/(2+3*x)^5+363/9604/(2+3*x) 
^4-1089/16807/(2+3*x)^3-7260/117649/(2+3*x)^2-45012/823543/(2+3*x)-17424/8 
23543*ln(1-2*x)+17424/823543*ln(2+3*x)
 

3.16.86.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^7} \, dx=\frac {4 \left (-\frac {7 \left (-145404842-461259404 x+887377581 x^2+5935583610 x^3+10278112680 x^4+7811789040 x^5+2286377280 x^6\right )}{16 (-1+2 x) (2+3 x)^6}-588060 \log (1-2 x)+588060 \log (4+6 x)\right )}{111178305} \]

input

Integrate[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^7),x]
 

output

(4*((-7*(-145404842 - 461259404*x + 887377581*x^2 + 5935583610*x^3 + 10278 
112680*x^4 + 7811789040*x^5 + 2286377280*x^6))/(16*(-1 + 2*x)*(2 + 3*x)^6) 
 - 588060*Log[1 - 2*x] + 588060*Log[4 + 6*x]))/111178305
 

3.16.86.3 Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 98, 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.

\(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^2 (3 x+2)^7} \, dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {52272}{823543 (3 x+2)}+\frac {135036}{823543 (3 x+2)^2}+\frac {43560}{117649 (3 x+2)^3}+\frac {9801}{16807 (3 x+2)^4}-\frac {1089}{2401 (3 x+2)^5}+\frac {101}{1029 (3 x+2)^6}-\frac {1}{147 (3 x+2)^7}-\frac {34848}{823543 (2 x-1)}+\frac {21296}{823543 (2 x-1)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {10648}{823543 (1-2 x)}-\frac {45012}{823543 (3 x+2)}-\frac {7260}{117649 (3 x+2)^2}-\frac {1089}{16807 (3 x+2)^3}+\frac {363}{9604 (3 x+2)^4}-\frac {101}{15435 (3 x+2)^5}+\frac {1}{2646 (3 x+2)^6}-\frac {17424 \log (1-2 x)}{823543}+\frac {17424 \log (3 x+2)}{823543}\)

input

Int[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^7),x]
 

output

10648/(823543*(1 - 2*x)) + 1/(2646*(2 + 3*x)^6) - 101/(15435*(2 + 3*x)^5) 
+ 363/(9604*(2 + 3*x)^4) - 1089/(16807*(2 + 3*x)^3) - 7260/(117649*(2 + 3* 
x)^2) - 45012/(823543*(2 + 3*x)) - (17424*Log[1 - 2*x])/823543 + (17424*Lo 
g[2 + 3*x])/823543
 

3.16.86.3.1 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]
 

3.16.86.4 Maple [A] (verified)

Time = 2.69 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.64

method	result	size
norman	\(\frac {-\frac {98597509}{7058940} x^{2}-\frac {19033542}{117649} x^{4}-\frac {14466276}{117649} x^{5}-\frac {4234032}{117649} x^{6}+\frac {115314851}{15882615} x -\frac {21983643}{235298} x^{3}+\frac {72702421}{31765230}}{\left (-1+2 x \right ) \left (2+3 x \right )^{6}}-\frac {17424 \ln \left (-1+2 x \right )}{823543}+\frac {17424 \ln \left (2+3 x \right )}{823543}\)	\(63\)
risch	\(\frac {-\frac {98597509}{7058940} x^{2}-\frac {19033542}{117649} x^{4}-\frac {14466276}{117649} x^{5}-\frac {4234032}{117649} x^{6}+\frac {115314851}{15882615} x -\frac {21983643}{235298} x^{3}+\frac {72702421}{31765230}}{\left (-1+2 x \right ) \left (2+3 x \right )^{6}}-\frac {17424 \ln \left (-1+2 x \right )}{823543}+\frac {17424 \ln \left (2+3 x \right )}{823543}\)	\(64\)
default	\(-\frac {10648}{823543 \left (-1+2 x \right )}-\frac {17424 \ln \left (-1+2 x \right )}{823543}+\frac {1}{2646 \left (2+3 x \right )^{6}}-\frac {101}{15435 \left (2+3 x \right )^{5}}+\frac {363}{9604 \left (2+3 x \right )^{4}}-\frac {1089}{16807 \left (2+3 x \right )^{3}}-\frac {7260}{117649 \left (2+3 x \right )^{2}}-\frac {45012}{823543 \left (2+3 x \right )}+\frac {17424 \ln \left (2+3 x \right )}{823543}\)	\(81\)
parallelrisch	\(\frac {-4617506880 x -11240570880 \ln \left (\frac {2}{3}+x \right ) x^{2}-4995809280 \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}-49243360320 x^{3}-14021895580 x^{4}-26361513360 x^{2}-42152140800 \ln \left (x -\frac {1}{2}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}-713687040 \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{7}[phone] \ln \left (x -\frac {1}{2}\right ) x^{2}[phone] \ln \left (x -\frac {1}{2}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}[phone] \ln \left (x -\frac {1}{2}\right )-16258682880 \ln \left (x -\frac {1}{2}\right ) x^{7}-56905390080 \ln \left (x -\frac {1}{2}\right ) x^{6}-75873853440 \ln \left (x -\frac {1}{2}\right ) x^{5}}{527067520 \left (-1+2 x \right ) \left (2+3 x \right )^{6}}\)	\(167\)

input

int((3+5*x)^3/(1-2*x)^2/(2+3*x)^7,x,method=_RETURNVERBOSE)
 

output

(-98597509/7058940*x^2-19033542/117649*x^4-14466276/117649*x^5-4234032/117 
649*x^6+115314851/15882615*x-21983643/235298*x^3+72702421/31765230)/(-1+2* 
x)/(2+3*x)^6-17424/823543*ln(-1+2*x)+17424/823543*ln(2+3*x)
 

3.16.86.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.43 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^7} \, dx=-\frac {16004640960 \, x^{6} + 54682523280 \, x^{5} + 71946788760 \, x^{4} + 41549085270 \, x^{3} + 6211643067 \, x^{2} - 9408960 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (3 \, x + 2\right ) + 9408960 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (2 \, x - 1\right ) - 3228815828 \, x - 1017833894}{444713220 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} \]

input

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^7,x, algorithm="fricas")
 

output

-1/444713220*(16004640960*x^6 + 54682523280*x^5 + 71946788760*x^4 + 415490 
85270*x^3 + 6211643067*x^2 - 9408960*(1458*x^7 + 5103*x^6 + 6804*x^5 + 378 
0*x^4 - 1008*x^2 - 448*x - 64)*log(3*x + 2) + 9408960*(1458*x^7 + 5103*x^6 
 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)*log(2*x - 1) - 3228815828* 
x - 1017833894)/(1458*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 44 
8*x - 64)
 

3.16.86.6 Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^7} \, dx=\frac {- 2286377280 x^{6} - 7811789040 x^{5} - 10278112680 x^{4} - 5935583610 x^{3} - 887377581 x^{2} + 461259404 x + 145404842}{92627410680 x^{7} + 324195937380 x^{6} + 432261249840 x^{5} + 240145138800 x^{4} - 64038703680 x^{2} - 28461646080 x - 4065949440} - \frac {17424 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {17424 \log {\left (x + \frac {2}{3} \right )}}{823543} \]

input

integrate((3+5*x)**3/(1-2*x)**2/(2+3*x)**7,x)
 

output

(-2286377280*x**6 - 7811789040*x**5 - 10278112680*x**4 - 5935583610*x**3 - 
 887377581*x**2 + 461259404*x + 145404842)/(92627410680*x**7 + 32419593738 
0*x**6 + 432261249840*x**5 + 240145138800*x**4 - 64038703680*x**2 - 284616 
46080*x - 4065949440) - 17424*log(x - 1/2)/823543 + 17424*log(x + 2/3)/823 
543
 

3.16.86.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.83 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^7} \, dx=-\frac {2286377280 \, x^{6} + 7811789040 \, x^{5} + 10278112680 \, x^{4} + 5935583610 \, x^{3} + 887377581 \, x^{2} - 461259404 \, x - 145404842}{63530460 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} + \frac {17424}{823543} \, \log \left (3 \, x + 2\right ) - \frac {17424}{823543} \, \log \left (2 \, x - 1\right ) \]

input

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^7,x, algorithm="maxima")
 

output

-1/63530460*(2286377280*x^6 + 7811789040*x^5 + 10278112680*x^4 + 593558361 
0*x^3 + 887377581*x^2 - 461259404*x - 145404842)/(1458*x^7 + 5103*x^6 + 68 
04*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64) + 17424/823543*log(3*x + 2) - 1 
7424/823543*log(2*x - 1)
 

3.16.86.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^7} \, dx=-\frac {10648}{823543 \, {\left (2 \, x - 1\right )}} + \frac {4 \, {\left (\frac {1421066052}{2 \, x - 1} + \frac {7028898345}{{\left (2 \, x - 1\right )}^{2}} + \frac {17396565550}{{\left (2 \, x - 1\right )}^{3}} + \frac {21521363500}{{\left (2 \, x - 1\right )}^{4}} + \frac {10637822580}{{\left (2 \, x - 1\right )}^{5}} + 115177113\right )}}{28824005 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{6}} + \frac {17424}{823543} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]

input

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^7,x, algorithm="giac")
 

output

-10648/823543/(2*x - 1) + 4/28824005*(1421066052/(2*x - 1) + 7028898345/(2 
*x - 1)^2 + 17396565550/(2*x - 1)^3 + 21521363500/(2*x - 1)^4 + 1063782258 
0/(2*x - 1)^5 + 115177113)/(7/(2*x - 1) + 3)^6 + 17424/823543*log(abs(-7/( 
2*x - 1) - 3))
 

3.16.86.9 Mupad [B] (verification not implemented)

Time = 1.48 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^7} \, dx=\frac {34848\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {2904\,x^6}{117649}+\frac {9922\,x^5}{117649}+\frac {117491\,x^4}{1058841}+\frac {271403\,x^3}{4235364}+\frac {98597509\,x^2}{10291934520}-\frac {115314851\,x}{23156852670}-\frac {72702421}{46313705340}}{x^7+\frac {7\,x^6}{2}+\frac {14\,x^5}{3}+\frac {70\,x^4}{27}-\frac {56\,x^2}{81}-\frac {224\,x}{729}-\frac {32}{729}} \]

input

int((5*x + 3)^3/((2*x - 1)^2*(3*x + 2)^7),x)
 

output

(34848*atanh((12*x)/7 + 1/7))/823543 - ((98597509*x^2)/10291934520 - (1153 
14851*x)/23156852670 + (271403*x^3)/4235364 + (117491*x^4)/1058841 + (9922 
*x^5)/117649 + (2904*x^6)/117649 - 72702421/46313705340)/((70*x^4)/27 - (5 
6*x^2)/81 - (224*x)/729 + (14*x^5)/3 + (7*x^6)/2 + x^7 - 32/729)