3.15.85 \(\int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx\) [1485]

3.15.85.1 Optimal result
3.15.85.2 Mathematica [A] (verified)
3.15.85.3 Rubi [A] (verified)
3.15.85.4 Maple [A] (verified)
3.15.85.5 Fricas [A] (verification not implemented)
3.15.85.6 Sympy [A] (verification not implemented)
3.15.85.7 Maxima [A] (verification not implemented)
3.15.85.8 Giac [A] (verification not implemented)
3.15.85.9 Mupad [B] (verification not implemented)

3.15.85.1 Optimal result

Integrand size = 22, antiderivative size = 68 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {40089855591 x}{10000000}-\frac {7136193339 x^2}{2000000}-\frac {345533877 x^3}{100000}-\frac {111146499 x^4}{40000}-\frac {8018271 x^5}{5000}-\frac {114453 x^6}{200}-\frac {6561 x^7}{70}-\frac {5764801 \log (1-2 x)}{2816}+\frac {\log (3+5 x)}{4296875} \]

output
-40089855591/10000000*x-7136193339/2000000*x^2-345533877/100000*x^3-111146 
499/40000*x^4-8018271/5000*x^5-114453/200*x^6-6561/70*x^7-5764801/2816*ln( 
1-2*x)+1/4296875*ln(3+5*x)
 
3.15.85.2 Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {3 \left (40324556806+93542996379 x+83255588955 x^2+80624571300 x^3+64835457750 x^4+37418598000 x^5+13352850000 x^6+2187000000 x^7\right )}{70000000}-\frac {5764801 \log (3-6 x)}{2816}+\frac {\log (-3 (3+5 x))}{4296875} \]

input
Integrate[(2 + 3*x)^8/((1 - 2*x)*(3 + 5*x)),x]
 
output
(-3*(40324556806 + 93542996379*x + 83255588955*x^2 + 80624571300*x^3 + 648 
35457750*x^4 + 37418598000*x^5 + 13352850000*x^6 + 2187000000*x^7))/700000 
00 - (5764801*Log[3 - 6*x])/2816 + Log[-3*(3 + 5*x)]/4296875
 
3.15.85.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 68, 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 = {93, 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.

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

\(\Big \downarrow \) 93

\(\displaystyle \int \left (-\frac {6561 x^6}{10}-\frac {343359 x^5}{100}-\frac {8018271 x^4}{1000}-\frac {111146499 x^3}{10000}-\frac {1036601631 x^2}{100000}-\frac {7136193339 x}{1000000}-\frac {5764801}{1408 (2 x-1)}+\frac {1}{859375 (5 x+3)}-\frac {40089855591}{10000000}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6561 x^7}{70}-\frac {114453 x^6}{200}-\frac {8018271 x^5}{5000}-\frac {111146499 x^4}{40000}-\frac {345533877 x^3}{100000}-\frac {7136193339 x^2}{2000000}-\frac {40089855591 x}{10000000}-\frac {5764801 \log (1-2 x)}{2816}+\frac {\log (5 x+3)}{4296875}\)

input
Int[(2 + 3*x)^8/((1 - 2*x)*(3 + 5*x)),x]
 
output
(-40089855591*x)/10000000 - (7136193339*x^2)/2000000 - (345533877*x^3)/100 
000 - (111146499*x^4)/40000 - (8018271*x^5)/5000 - (114453*x^6)/200 - (656 
1*x^7)/70 - (5764801*Log[1 - 2*x])/2816 + Log[3 + 5*x]/4296875
 

3.15.85.3.1 Defintions of rubi rules used

rule 93
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre 
eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
 

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 
3.15.85.4 Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.69

method result size
parallelrisch \(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (x +\frac {3}{5}\right )}{4296875}-\frac {5764801 \ln \left (x -\frac {1}{2}\right )}{2816}\) \(47\)
default \(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (3+5 x \right )}{4296875}-\frac {5764801 \ln \left (-1+2 x \right )}{2816}\) \(51\)
norman \(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (3+5 x \right )}{4296875}-\frac {5764801 \ln \left (-1+2 x \right )}{2816}\) \(51\)
risch \(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (3+5 x \right )}{4296875}-\frac {5764801 \ln \left (-1+2 x \right )}{2816}\) \(51\)

input
int((2+3*x)^8/(1-2*x)/(3+5*x),x,method=_RETURNVERBOSE)
 
output
-6561/70*x^7-114453/200*x^6-8018271/5000*x^5-111146499/40000*x^4-345533877 
/100000*x^3-7136193339/2000000*x^2-40089855591/10000000*x+1/4296875*ln(x+3 
/5)-5764801/2816*ln(x-1/2)
 
3.15.85.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {6561}{70} \, x^{7} - \frac {114453}{200} \, x^{6} - \frac {8018271}{5000} \, x^{5} - \frac {111146499}{40000} \, x^{4} - \frac {345533877}{100000} \, x^{3} - \frac {7136193339}{2000000} \, x^{2} - \frac {40089855591}{10000000} \, x + \frac {1}{4296875} \, \log \left (5 \, x + 3\right ) - \frac {5764801}{2816} \, \log \left (2 \, x - 1\right ) \]

input
integrate((2+3*x)^8/(1-2*x)/(3+5*x),x, algorithm="fricas")
 
output
-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 3 
45533877/100000*x^3 - 7136193339/2000000*x^2 - 40089855591/10000000*x + 1/ 
4296875*log(5*x + 3) - 5764801/2816*log(2*x - 1)
 
3.15.85.6 Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=- \frac {6561 x^{7}}{70} - \frac {114453 x^{6}}{200} - \frac {8018271 x^{5}}{5000} - \frac {111146499 x^{4}}{40000} - \frac {345533877 x^{3}}{100000} - \frac {7136193339 x^{2}}{2000000} - \frac {40089855591 x}{10000000} - \frac {5764801 \log {\left (x - \frac {1}{2} \right )}}{2816} + \frac {\log {\left (x + \frac {3}{5} \right )}}{4296875} \]

input
integrate((2+3*x)**8/(1-2*x)/(3+5*x),x)
 
output
-6561*x**7/70 - 114453*x**6/200 - 8018271*x**5/5000 - 111146499*x**4/40000 
 - 345533877*x**3/100000 - 7136193339*x**2/2000000 - 40089855591*x/1000000 
0 - 5764801*log(x - 1/2)/2816 + log(x + 3/5)/4296875
 
3.15.85.7 Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {6561}{70} \, x^{7} - \frac {114453}{200} \, x^{6} - \frac {8018271}{5000} \, x^{5} - \frac {111146499}{40000} \, x^{4} - \frac {345533877}{100000} \, x^{3} - \frac {7136193339}{2000000} \, x^{2} - \frac {40089855591}{10000000} \, x + \frac {1}{4296875} \, \log \left (5 \, x + 3\right ) - \frac {5764801}{2816} \, \log \left (2 \, x - 1\right ) \]

input
integrate((2+3*x)^8/(1-2*x)/(3+5*x),x, algorithm="maxima")
 
output
-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 3 
45533877/100000*x^3 - 7136193339/2000000*x^2 - 40089855591/10000000*x + 1/ 
4296875*log(5*x + 3) - 5764801/2816*log(2*x - 1)
 
3.15.85.8 Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {6561}{70} \, x^{7} - \frac {114453}{200} \, x^{6} - \frac {8018271}{5000} \, x^{5} - \frac {111146499}{40000} \, x^{4} - \frac {345533877}{100000} \, x^{3} - \frac {7136193339}{2000000} \, x^{2} - \frac {40089855591}{10000000} \, x + \frac {1}{4296875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {5764801}{2816} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

input
integrate((2+3*x)^8/(1-2*x)/(3+5*x),x, algorithm="giac")
 
output
-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 3 
45533877/100000*x^3 - 7136193339/2000000*x^2 - 40089855591/10000000*x + 1/ 
4296875*log(abs(5*x + 3)) - 5764801/2816*log(abs(2*x - 1))
 
3.15.85.9 Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=\frac {\ln \left (x+\frac {3}{5}\right )}{4296875}-\frac {5764801\,\ln \left (x-\frac {1}{2}\right )}{2816}-\frac {40089855591\,x}{10000000}-\frac {7136193339\,x^2}{2000000}-\frac {345533877\,x^3}{100000}-\frac {111146499\,x^4}{40000}-\frac {8018271\,x^5}{5000}-\frac {114453\,x^6}{200}-\frac {6561\,x^7}{70} \]

input
int(-(3*x + 2)^8/((2*x - 1)*(5*x + 3)),x)
 
output
log(x + 3/5)/4296875 - (5764801*log(x - 1/2))/2816 - (40089855591*x)/10000 
000 - (7136193339*x^2)/2000000 - (345533877*x^3)/100000 - (111146499*x^4)/ 
40000 - (8018271*x^5)/5000 - (114453*x^6)/200 - (6561*x^7)/70