\(\int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx\) [462]

Optimal result

Integrand size = 22, antiderivative size = 84 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {5764801}{170368 (1-2 x)^2}-\frac {130943337}{937024 (1-2 x)}-\frac {242028 x}{3125}-\frac {330237 x^2}{20000}-\frac {2187 x^3}{1000}-\frac {1}{41593750 (3+5 x)^2}-\frac {54}{45753125 (3+5 x)}-\frac {595421589 \log (1-2 x)}{5153632}+\frac {1284 \log (3+5 x)}{100656875} \] Output:

5764801/170368/(1-2*x)^2-130943337/(937024-1874048*x)-242028/3125*x-330237 
/20000*x^2-2187/1000*x^3-1/41593750/(3+5*x)^2-54/(137259375+228765625*x)-5 
95421589/5153632*ln(1-2*x)+1284/100656875*ln(3+5*x)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {-\frac {11 \left (39754322426279+31893783102814 x-254889143270829 x^2-314407515766380 x^3+148045752548100 x^4+232677700200000 x^5+49630793850000 x^6+6403973400000 x^7\right )}{\left (-3+x+10 x^2\right )^2}-37213849312500 \log (3-6 x)+4108800 \log (-3 (3+5 x))}{322102000000} \] Input:

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

Output:

((-11*(39754322426279 + 31893783102814*x - 254889143270829*x^2 - 314407515 
766380*x^3 + 148045752548100*x^4 + 232677700200000*x^5 + 49630793850000*x^ 
6 + 6403973400000*x^7))/(-3 + x + 10*x^2)^2 - 37213849312500*Log[3 - 6*x] 
+ 4108800*Log[-3*(3 + 5*x)])/322102000000
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 84, 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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {6561 x^2}{1000}-\frac {330237 x}{10000}-\frac {595421589}{2576816 (2 x-1)}+\frac {1284}{20131375 (5 x+3)}-\frac {130943337}{468512 (2 x-1)^2}+\frac {54}{9150625 (5 x+3)^2}-\frac {5764801}{42592 (2 x-1)^3}+\frac {1}{4159375 (5 x+3)^3}-\frac {242028}{3125}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2187 x^3}{1000}-\frac {330237 x^2}{20000}-\frac {242028 x}{3125}-\frac {130943337}{937024 (1-2 x)}-\frac {54}{45753125 (5 x+3)}+\frac {5764801}{170368 (1-2 x)^2}-\frac {1}{41593750 (5 x+3)^2}-\frac {595421589 \log (1-2 x)}{5153632}+\frac {1284 \log (5 x+3)}{100656875}\)

Input:

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

Output:

5764801/(170368*(1 - 2*x)^2) - 130943337/(937024*(1 - 2*x)) - (242028*x)/3 
125 - (330237*x^2)/20000 - (2187*x^3)/1000 - 1/(41593750*(3 + 5*x)^2) - 54 
/(45753125*(3 + 5*x)) - (595421589*Log[1 - 2*x])/5153632 + (1284*Log[3 + 5 
*x])/100656875
 

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]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {2187 x^{3}}{1000}-\frac {330237 x^{2}}{20000}-\frac {242028 x}{3125}+\frac {\frac {2045989633713}{292820000} x^{3}+\frac {167989904414289}{29282000000} x^{2}-\frac {9689497987007}{14641000000} x -\frac {27910387088759}{29282000000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {595421589 \ln \left (-1+2 x \right )}{5153632}+\frac {1284 \ln \left (3+5 x \right )}{100656875}\)	\(62\)
norman	\(\frac {-\frac {407765282557}{292820000} x +\frac {86004166569}{7320500} x^{3}+\frac {3350842985349}{585640000} x^{2}-\frac {79461}{10} x^{5}-\frac {67797}{40} x^{6}-\frac {2187}{10} x^{7}-\frac {528604093939}{585640000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {595421589 \ln \left (-1+2 x \right )}{5153632}+\frac {1284 \ln \left (3+5 x \right )}{100656875}\)	\(63\)
default	\(-\frac {2187 x^{3}}{1000}-\frac {330237 x^{2}}{20000}-\frac {242028 x}{3125}-\frac {1}{41593750 \left (3+5 x \right )^{2}}-\frac {54}{45753125 \left (3+5 x \right )}+\frac {1284 \ln \left (3+5 x \right )}{100656875}+\frac {5764801}{170368 \left (-1+2 x \right )^{2}}+\frac {130943337}{937024 \left (-1+2 x \right )}-\frac {595421589 \ln \left (-1+2 x \right )}{5153632}\)	\(67\)
parallelrisch	\(-\frac {2113311222000 x^{7}[phone] x^{6}[phone] \ln \left (x -\frac {1}{2}\right ) x^{4}-12326400 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] x^{5}[phone]+22328309587500 \ln \left (x -\frac {1}{2}\right ) x^{3}-2465280 \ln \left (x +\frac {3}{5}\right ) x^{3}[phone] x^{4}-65868513283125 \ln \left (x -\frac {1}{2}\right ) x^{2}+7272576 \ln \left (x +\frac {3}{5}\right ) x^{2}-68671318789810 x^{3}-6698492876250 \ln \left (x -\frac {1}{2}\right ) x +739584 \ln \left (x +\frac {3}{5}\right ) x -187608743448005 x^{2}[phone] \ln \left (x -\frac {1}{2}\right )-1109376 \ln \left (x +\frac {3}{5}\right )}{9663060000 \left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}\)	\(129\)

Input:

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

Output:

-2187/1000*x^3-330237/20000*x^2-242028/3125*x+100*(2045989633713/292820000 
00*x^3+167989904414289/2928200000000*x^2-9689497987007/1464100000000*x-279 
10387088759/2928200000000)/(-1+2*x)^2/(3+5*x)^2-595421589/5153632*ln(-1+2* 
x)+1284/100656875*ln(3+5*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.37 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {70443707400000 \, x^{7} + 545938732350000 \, x^{6} + 2559454702200000 \, x^{5} + 180911181221100 \, x^{4} - 3748001092791780 \, x^{3} - 1949701238862399 \, x^{2} - 4108800 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 37213849312500 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 437687139939434 \, x + 307014257976349}{322102000000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \] Input:

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

Output:

-1/322102000000*(70443707400000*x^7 + 545938732350000*x^6 + 25594547022000 
00*x^5 + 180911181221100*x^4 - 3748001092791780*x^3 - 1949701238862399*x^2 
 - 4108800*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) + 3721384931 
2500*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) + 437687139939434* 
x + 307014257976349)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=- \frac {2187 x^{3}}{1000} - \frac {330237 x^{2}}{20000} - \frac {242028 x}{3125} - \frac {- 204598963371300 x^{3} - 167989904414289 x^{2} + 19378995974014 x + 27910387088759}{2928200000000 x^{4} + 585640000000 x^{3} - 1727638000000 x^{2} - 175692000000 x + 263538000000} - \frac {595421589 \log {\left (x - \frac {1}{2} \right )}}{5153632} + \frac {1284 \log {\left (x + \frac {3}{5} \right )}}{100656875} \] Input:

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

Output:

-2187*x**3/1000 - 330237*x**2/20000 - 242028*x/3125 - (-204598963371300*x* 
*3 - 167989904414289*x**2 + 19378995974014*x + 27910387088759)/(2928200000 
000*x**4 + 585640000000*x**3 - 1727638000000*x**2 - 175692000000*x + 26353 
8000000) - 595421589*log(x - 1/2)/5153632 + 1284*log(x + 3/5)/100656875
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.82 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {2187}{1000} \, x^{3} - \frac {330237}{20000} \, x^{2} - \frac {242028}{3125} \, x + \frac {204598963371300 \, x^{3} + 167989904414289 \, x^{2} - 19378995974014 \, x - 27910387088759}{29282000000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {1284}{100656875} \, \log \left (5 \, x + 3\right ) - \frac {595421589}{5153632} \, \log \left (2 \, x - 1\right ) \] Input:

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

Output:

-2187/1000*x^3 - 330237/20000*x^2 - 242028/3125*x + 1/29282000000*(2045989 
63371300*x^3 + 167989904414289*x^2 - 19378995974014*x - 27910387088759)/(1 
00*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 1284/100656875*log(5*x + 3) - 595421 
589/5153632*log(2*x - 1)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {2187}{1000} \, x^{3} - \frac {330237}{20000} \, x^{2} - \frac {242028}{3125} \, x + \frac {204598963371300 \, x^{3} + 167989904414289 \, x^{2} - 19378995974014 \, x - 27910387088759}{29282000000 \, {\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {1284}{100656875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {595421589}{5153632} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \] Input:

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

Output:

-2187/1000*x^3 - 330237/20000*x^2 - 242028/3125*x + 1/29282000000*(2045989 
63371300*x^3 + 167989904414289*x^2 - 19378995974014*x - 27910387088759)/(( 
5*x + 3)^2*(2*x - 1)^2) + 1284/100656875*log(abs(5*x + 3)) - 595421589/515 
3632*log(abs(2*x - 1))
 

Mupad [B] (verification not implemented)

Time = 1.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {1284\,\ln \left (x+\frac {3}{5}\right )}{100656875}-\frac {595421589\,\ln \left (x-\frac {1}{2}\right )}{5153632}-\frac {242028\,x}{3125}-\frac {-\frac {2045989633713\,x^3}{29282000000}-\frac {167989904414289\,x^2}{2928200000000}+\frac {9689497987007\,x}{1464100000000}+\frac {27910387088759}{2928200000000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}}-\frac {330237\,x^2}{20000}-\frac {2187\,x^3}{1000} \] Input:

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

Output:

(1284*log(x + 3/5))/100656875 - (595421589*log(x - 1/2))/5153632 - (242028 
*x)/3125 - ((9689497987007*x)/1464100000000 - (167989904414289*x^2)/292820 
0000000 - (2045989633713*x^3)/29282000000 + 27910387088759/2928200000000)/ 
(x^3/5 - (59*x^2)/100 - (3*x)/50 + x^4 + 9/100) - (330237*x^2)/20000 - (21 
87*x^3)/1000
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.82 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {8217600 \,\mathrm {log}\left (5 x +3\right ) x^{4}+1643520 \,\mathrm {log}\left (5 x +3\right ) x^{3}-4848384 \,\mathrm {log}\left (5 x +3\right ) x^{2}-493056 \,\mathrm {log}\left (5 x +3\right ) x +739584 \,\mathrm {log}\left (5 x +3\right )-74427698625000 \,\mathrm {log}\left (2 x -1\right ) x^{4}-14885539725000 \,\mathrm {log}\left (2 x -1\right ) x^{3}[phone] \,\mathrm {log}\left (2 x -1\right ) x^{2}[phone] \,\mathrm {log}\left (2 x -1\right ) x -6698492876250 \,\mathrm {log}\left (2 x -1\right )-1408874148000 x^{7}-10918774647000 x^{6}-51189094044000 x^{5}-378418332903600 x^{4}[phone] x^{2}[phone] x -39872294994653}{644204000000 x^{4}[phone] x^{3}-380080360000 x^{2}-38652240000 x [phone]} \] Input:

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

Output:

(8217600*log(5*x + 3)*x**4 + 1643520*log(5*x + 3)*x**3 - 4848384*log(5*x + 
 3)*x**2 - 493056*log(5*x + 3)*x + 739584*log(5*x + 3) - 74427698625000*lo 
g(2*x - 1)*x**4 - 14885539725000*log(2*x - 1)*x**3 + 43912342188750*log(2* 
x - 1)*x**2 + 4465661917500*log(2*x - 1)*x - 6698492876250*log(2*x - 1) - 
1408874148000*x**7 - 10918774647000*x**6 - 51189094044000*x**5 - 378418332 
903600*x**4 + 260126089251963*x**2 + 13734263757962*x - 39872294994653)/(6 
442040000*(100*x**4 + 20*x**3 - 59*x**2 - 6*x + 9))