3.17.84 \(\int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^2} \, dx\) [1684]

3.17.84.1 Optimal result

Integrand size = 22, antiderivative size = 80 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {5764801}{30976 (1-2 x)^2}-\frac {79883671}{85184 (1-2 x)}-\frac {81001863 x}{100000}-\frac {4863159 x^2}{20000}-\frac {123201 x^3}{2000}-\frac {6561 x^4}{800}-\frac {1}{20796875 (3+5 x)}-\frac {1845559863 \log (1-2 x)}{1874048}+\frac {54 \log (3+5 x)}{45753125} \]

output

5764801/30976/(1-2*x)^2-79883671/85184/(1-2*x)-81001863/100000*x-4863159/2 
0000*x^2-123201/2000*x^3-6561/800*x^4-1/20796875/(3+5*x)-1845559863/187404 
8*ln(1-2*x)+54/45753125*ln(3+5*x)
 

3.17.84.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.22 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {5764801}{30976 (1-2 x)^2}+\frac {79883671}{85184 (-1+2 x)}-\frac {18607401 (2+3 x)}{100000}-\frac {315171 (2+3 x)^2}{20000}-\frac {2943 (2+3 x)^3}{2000}-\frac {81}{800} (2+3 x)^4-\frac {1}{20796875 (3+5 x)}-\frac {1845559863 \log (3-6 x)}{1874048}+\frac {54 \log (-3 (3+5 x))}{45753125} \]

input

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

output

5764801/(30976*(1 - 2*x)^2) + 79883671/(85184*(-1 + 2*x)) - (18607401*(2 + 
 3*x))/100000 - (315171*(2 + 3*x)^2)/20000 - (2943*(2 + 3*x)^3)/2000 - (81 
*(2 + 3*x)^4)/800 - 1/(20796875*(3 + 5*x)) - (1845559863*Log[3 - 6*x])/187 
4048 + (54*Log[-3*(3 + 5*x)])/45753125
 

3.17.84.3 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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {6561 x^3}{200}-\frac {369603 x^2}{2000}-\frac {4863159 x}{10000}-\frac {1845559863}{937024 (2 x-1)}+\frac {54}{9150625 (5 x+3)}-\frac {79883671}{42592 (2 x-1)^2}+\frac {1}{4159375 (5 x+3)^2}-\frac {5764801}{7744 (2 x-1)^3}-\frac {81001863}{100000}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6561 x^4}{800}-\frac {123201 x^3}{2000}-\frac {4863159 x^2}{20000}-\frac {81001863 x}{100000}-\frac {79883671}{85184 (1-2 x)}-\frac {1}{20796875 (5 x+3)}+\frac {5764801}{30976 (1-2 x)^2}-\frac {1845559863 \log (1-2 x)}{1874048}+\frac {54 \log (5 x+3)}{45753125}\)

input

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

output

5764801/(30976*(1 - 2*x)^2) - 79883671/(85184*(1 - 2*x)) - (81001863*x)/10 
0000 - (4863159*x^2)/20000 - (123201*x^3)/2000 - (6561*x^4)/800 - 1/(20796 
875*(3 + 5*x)) - (1845559863*Log[1 - 2*x])/1874048 + (54*Log[3 + 5*x])/457 
53125
 

3.17.84.3.1 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]
 

3.17.84.4 Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78

method	result	size
risch	\(-\frac {6561 x^{4}}{800}-\frac {123201 x^{3}}{2000}-\frac {4863159 x^{2}}{20000}-\frac {81001863 x}{100000}+\frac {\frac {6240911796747}{665500000} x^{2}+\frac {9946855297899}{5324000000} x -\frac {12005712797131}{5324000000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {1845559863 \ln \left (-1+2 x \right )}{1874048}+\frac {54 \ln \left (3+5 x \right )}{45753125}\)	\(62\)
default	\(-\frac {6561 x^{4}}{800}-\frac {123201 x^{3}}{2000}-\frac {4863159 x^{2}}{20000}-\frac {81001863 x}{100000}-\frac {1}{20796875 \left (3+5 x \right )}+\frac {54 \ln \left (3+5 x \right )}{45753125}+\frac {5764801}{30976 \left (-1+2 x \right )^{2}}+\frac {79883671}{85184 \left (-1+2 x \right )}-\frac {1845559863 \ln \left (-1+2 x \right )}{1874048}\)	\(63\)
norman	\(\frac {\frac {331620868303}{39930000} x^{2}-\frac {930122763613}{159720000} x +\frac {3678495782933}{159720000} x^{3}-\frac {55394037}{4000} x^{4}-\frac {3450357}{800} x^{5}-\frac {5832}{5} x^{6}-\frac {6561}{40} x^{7}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {1845559863 \ln \left (-1+2 x \right )}{1874048}+\frac {54 \ln \left (3+5 x \right )}{45753125}\)	\(67\)
parallelrisch	\(\frac {-2881788030000 x^{7}-20492714880000 x^{6}-75775015255500 x^{5}+414720 \ln \left (x +\frac {3}{5}\right ) x^{3}-346042474312500 \ln \left (x -\frac {1}{2}\right ) x^{3}-243307228715100 x^{4}-165888 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (x -\frac {1}{2}\right ) x^{2}[phone] x^{3}-145152 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (x -\frac {1}{2}\right ) x [phone] x^{2}+62208 \ln \left (x +\frac {3}{5}\right )-51906371146875 \ln \left (x -\frac {1}{2}\right )-102313503997430 x}{17569200000 \left (-1+2 x \right )^{2} \left (3+5 x \right )}\)	\(113\)

input

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

output

-6561/800*x^4-123201/2000*x^3-4863159/20000*x^2-81001863/100000*x+20*(6240 
911796747/13310000000*x^2+9946855297899/106480000000*x-12005712797131/1064 
80000000)/(-1+2*x)^2/(3+5*x)-1845559863/1874048*ln(-1+2*x)+54/45753125*ln( 
3+5*x)
 

3.17.84.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {9605960100000 \, x^{7} + 68309049600000 \, x^{6} + 252583384185000 \, x^{5} + 811024095717000 \, x^{4} - 468362848619160 \, x^{3} - 838544848893576 \, x^{2} - 69120 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 57673745718750 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 32898384865071 \, x + 132062840768441}{58564000000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]

input

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

output

-1/58564000000*(9605960100000*x^7 + 68309049600000*x^6 + 252583384185000*x 
^5 + 811024095717000*x^4 - 468362848619160*x^3 - 838544848893576*x^2 - 691 
20*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) + 57673745718750*(20*x^3 - 8*x^ 
2 - 7*x + 3)*log(2*x - 1) + 32898384865071*x + 132062840768441)/(20*x^3 - 
8*x^2 - 7*x + 3)
 

3.17.84.6 Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^2} \, dx=- \frac {6561 x^{4}}{800} - \frac {123201 x^{3}}{2000} - \frac {4863159 x^{2}}{20000} - \frac {81001863 x}{100000} - \frac {- 49927294373976 x^{2} - 9946855297899 x + 12005712797131}{106480000000 x^{3} - 42592000000 x^{2} - 37268000000 x + 15972000000} - \frac {1845559863 \log {\left (x - \frac {1}{2} \right )}}{1874048} + \frac {54 \log {\left (x + \frac {3}{5} \right )}}{45753125} \]

input

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

output

-6561*x**4/800 - 123201*x**3/2000 - 4863159*x**2/20000 - 81001863*x/100000 
 - (-49927294373976*x**2 - 9946855297899*x + 12005712797131)/(106480000000 
*x**3 - 42592000000*x**2 - 37268000000*x + 15972000000) - 1845559863*log(x 
 - 1/2)/1874048 + 54*log(x + 3/5)/45753125
 

3.17.84.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {6561}{800} \, x^{4} - \frac {123201}{2000} \, x^{3} - \frac {4863159}{20000} \, x^{2} - \frac {81001863}{100000} \, x + \frac {49927294373976 \, x^{2} + 9946855297899 \, x - 12005712797131}{5324000000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac {54}{45753125} \, \log \left (5 \, x + 3\right ) - \frac {1845559863}{1874048} \, \log \left (2 \, x - 1\right ) \]

input

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

output

-6561/800*x^4 - 123201/2000*x^3 - 4863159/20000*x^2 - 81001863/100000*x + 
1/5324000000*(49927294373976*x^2 + 9946855297899*x - 12005712797131)/(20*x 
^3 - 8*x^2 - 7*x + 3) + 54/45753125*log(5*x + 3) - 1845559863/1874048*log( 
2*x - 1)
 

3.17.84.8 Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.40 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {{\left (5 \, x + 3\right )}^{4} {\left (\frac {11185606872}{5 \, x + 3} + \frac {158583727962}{{\left (5 \, x + 3\right )}^{2}} + \frac {3495217526460}{{\left (5 \, x + 3\right )}^{3}} - \frac {86510680819405}{{\left (5 \, x + 3\right )}^{4}} + \frac {317205578854725}{{\left (5 \, x + 3\right )}^{5}} + 768476808\right )}}{14641000000 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2}} - \frac {1}{20796875 \, {\left (5 \, x + 3\right )}} + \frac {393919443}{400000} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {1845559863}{1874048} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]

input

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

output

-1/14641000000*(5*x + 3)^4*(11185606872/(5*x + 3) + 158583727962/(5*x + 3) 
^2 + 3495217526460/(5*x + 3)^3 - 86510680819405/(5*x + 3)^4 + 317205578854 
725/(5*x + 3)^5 + 768476808)/(11/(5*x + 3) - 2)^2 - 1/20796875/(5*x + 3) + 
 393919443/400000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 1845559863/1874048*l 
og(abs(-11/(5*x + 3) + 2))
 

3.17.84.9 Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {54\,\ln \left (x+\frac {3}{5}\right )}{45753125}-\frac {1845559863\,\ln \left (x-\frac {1}{2}\right )}{1874048}-\frac {81001863\,x}{100000}-\frac {\frac {6240911796747\,x^2}{13310000000}+\frac {9946855297899\,x}{106480000000}-\frac {12005712797131}{106480000000}}{-x^3+\frac {2\,x^2}{5}+\frac {7\,x}{20}-\frac {3}{20}}-\frac {4863159\,x^2}{20000}-\frac {123201\,x^3}{2000}-\frac {6561\,x^4}{800} \]

input

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

output

(54*log(x + 3/5))/45753125 - (1845559863*log(x - 1/2))/1874048 - (81001863 
*x)/100000 - ((9946855297899*x)/106480000000 + (6240911796747*x^2)/1331000 
0000 - 12005712797131/106480000000)/((7*x)/20 + (2*x^2)/5 - x^3 - 3/20) - 
(4863159*x^2)/20000 - (123201*x^3)/2000 - (6561*x^4)/800