\(\int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx\) [463]

Optimal result

Integrand size = 22, antiderivative size = 77 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {823543}{85184 (1-2 x)^2}-\frac {7411887}{234256 (1-2 x)}-\frac {95499 x}{10000}-\frac {2187 x^2}{2000}-\frac {1}{8318750 (3+5 x)^2}-\frac {237}{45753125 (3+5 x)}-\frac {25059237 \log (1-2 x)}{1288408}+\frac {24279 \log (3+5 x)}{503284375} \] Output:

823543/85184/(1-2*x)^2-7411887/(234256-468512*x)-95499/10000*x-2187/2000*x 
^2-1/8318750/(3+5*x)^2-237/(137259375+228765625*x)-25059237/1288408*ln(1-2 
*x)+24279/503284375*ln(3+5*x)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {-\frac {11 \left (734029874011+410862940766 x-5105353973121 x^2-5957126547060 x^3+2092320420300 x^4+2860441452000 x^5+320198670000 x^6\right )}{\left (-3+x+10 x^2\right )^2}-626480925000 \log (3-6 x)+1553856 \log (-3 (3+5 x))}{32210200000} \] Input:

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

Output:

((-11*(734029874011 + 410862940766*x - 5105353973121*x^2 - 5957126547060*x 
^3 + 2092320420300*x^4 + 2860441452000*x^5 + 320198670000*x^6))/(-3 + x + 
10*x^2)^2 - 626480925000*Log[3 - 6*x] + 1553856*Log[-3*(3 + 5*x)])/3221020 
0000
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 77, 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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {2187 x}{1000}-\frac {25059237}{644204 (2 x-1)}+\frac {24279}{100656875 (5 x+3)}-\frac {7411887}{117128 (2 x-1)^2}+\frac {237}{9150625 (5 x+3)^2}-\frac {823543}{21296 (2 x-1)^3}+\frac {1}{831875 (5 x+3)^3}-\frac {95499}{10000}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2187 x^2}{2000}-\frac {95499 x}{10000}-\frac {7411887}{234256 (1-2 x)}-\frac {237}{45753125 (5 x+3)}+\frac {823543}{85184 (1-2 x)^2}-\frac {1}{8318750 (5 x+3)^2}-\frac {25059237 \log (1-2 x)}{1288408}+\frac {24279 \log (5 x+3)}{503284375}\)

Input:

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

Output:

823543/(85184*(1 - 2*x)^2) - 7411887/(234256*(1 - 2*x)) - (95499*x)/10000 
- (2187*x^2)/2000 - 1/(8318750*(3 + 5*x)^2) - 237/(45753125*(3 + 5*x)) - ( 
25059237*Log[1 - 2*x])/1288408 + (24279*Log[3 + 5*x])/503284375
 

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 = 57, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {2187 x^{2}}{2000}-\frac {95499 x}{10000}+\frac {\frac {115810726791}{73205000} x^{3}+\frac {3950432948061}{2928200000} x^{2}-\frac {131252111833}{1464100000} x -\frac {579053717731}{2928200000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {25059237 \ln \left (-1+2 x \right )}{1288408}+\frac {24279 \ln \left (3+5 x \right )}{503284375}\)	\(57\)
norman	\(\frac {-\frac {16762567687}{91506250} x +\frac {79694882889}{36602500} x^{3}+\frac {483860615643}{366025000} x^{2}-\frac {48843}{50} x^{5}-\frac {2187}{20} x^{6}-\frac {68215129523}{366025000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {25059237 \ln \left (-1+2 x \right )}{1288408}+\frac {24279 \ln \left (3+5 x \right )}{503284375}\)	\(58\)
default	\(-\frac {2187 x^{2}}{2000}-\frac {95499 x}{10000}-\frac {1}{8318750 \left (3+5 x \right )^{2}}-\frac {237}{45753125 \left (3+5 x \right )}+\frac {24279 \ln \left (3+5 x \right )}{503284375}+\frac {823543}{85184 \left (-1+2 x \right )^{2}}+\frac {7411887}{234256 \left (-1+2 x \right )}-\frac {25059237 \ln \left (-1+2 x \right )}{1288408}\)	\(62\)
parallelrisch	\(-\frac {1320819513750 x^{6}[phone] \ln \left (x -\frac {1}{2}\right ) x^{4}-58269600 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] x^{5}[phone]+4698606937500 \ln \left (x -\frac {1}{2}\right ) x^{3}-11653920 \ln \left (x +\frac {3}{5}\right ) x^{3}[phone] x^{4}-13860890465625 \ln \left (x -\frac {1}{2}\right ) x^{2}[phone] \ln \left (x +\frac {3}{5}\right ) x^{2}-18923781571090 x^{3}-1409582081250 \ln \left (x -\frac {1}{2}\right ) x +3496176 \ln \left (x +\frac {3}{5}\right ) x -37725213173945 x^{2}[phone] \ln \left (x -\frac {1}{2}\right )-5244264 \ln \left (x +\frac {3}{5}\right )}{12078825000 \left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}\)	\(124\)

Input:

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

Output:

-2187/2000*x^2-95499/10000*x+100*(115810726791/7320500000*x^3+395043294806 
1/292820000000*x^2-131252111833/146410000000*x-579053717731/292820000000)/ 
(-1+2*x)^2/(3+5*x)^2-25059237/1288408*ln(-1+2*x)+24279/503284375*ln(3+5*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.43 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {3522185370000 \, x^{6} + 31464855972000 \, x^{5} + 4073994411300 \, x^{4} - 69316698060060 \, x^{3} - 44983390879251 \, x^{2} - 1553856 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 626480925000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 5655984161146 \, x + 6369590895041}{32210200000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \] Input:

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

Output:

-1/32210200000*(3522185370000*x^6 + 31464855972000*x^5 + 4073994411300*x^4 
 - 69316698060060*x^3 - 44983390879251*x^2 - 1553856*(100*x^4 + 20*x^3 - 5 
9*x^2 - 6*x + 9)*log(5*x + 3) + 626480925000*(100*x^4 + 20*x^3 - 59*x^2 - 
6*x + 9)*log(2*x - 1) + 5655984161146*x + 6369590895041)/(100*x^4 + 20*x^3 
 - 59*x^2 - 6*x + 9)
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=- \frac {2187 x^{2}}{2000} - \frac {95499 x}{10000} - \frac {- 4632429071640 x^{3} - 3950432948061 x^{2} + 262504223666 x + 579053717731}{292820000000 x^{4} + 58564000000 x^{3} - 172763800000 x^{2} - 17569200000 x + 26353800000} - \frac {25059237 \log {\left (x - \frac {1}{2} \right )}}{1288408} + \frac {24279 \log {\left (x + \frac {3}{5} \right )}}{503284375} \] Input:

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

Output:

-2187*x**2/2000 - 95499*x/10000 - (-4632429071640*x**3 - 3950432948061*x** 
2 + 262504223666*x + 579053717731)/(292820000000*x**4 + 58564000000*x**3 - 
 172763800000*x**2 - 17569200000*x + 26353800000) - 25059237*log(x - 1/2)/ 
1288408 + 24279*log(x + 3/5)/503284375
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.83 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {2187}{2000} \, x^{2} - \frac {95499}{10000} \, x + \frac {4632429071640 \, x^{3} + 3950432948061 \, x^{2} - 262504223666 \, x - 579053717731}{2928200000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {24279}{503284375} \, \log \left (5 \, x + 3\right ) - \frac {25059237}{1288408} \, \log \left (2 \, x - 1\right ) \] Input:

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

Output:

-2187/2000*x^2 - 95499/10000*x + 1/2928200000*(4632429071640*x^3 + 3950432 
948061*x^2 - 262504223666*x - 579053717731)/(100*x^4 + 20*x^3 - 59*x^2 - 6 
*x + 9) + 24279/503284375*log(5*x + 3) - 25059237/1288408*log(2*x - 1)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {2187}{2000} \, x^{2} - \frac {95499}{10000} \, x + \frac {4632429071640 \, x^{3} + 3950432948061 \, x^{2} - 262504223666 \, x - 579053717731}{2928200000 \, {\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {24279}{503284375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {25059237}{1288408} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \] Input:

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

Output:

-2187/2000*x^2 - 95499/10000*x + 1/2928200000*(4632429071640*x^3 + 3950432 
948061*x^2 - 262504223666*x - 579053717731)/((5*x + 3)^2*(2*x - 1)^2) + 24 
279/503284375*log(abs(5*x + 3)) - 25059237/1288408*log(abs(2*x - 1))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {24279\,\ln \left (x+\frac {3}{5}\right )}{503284375}-\frac {25059237\,\ln \left (x-\frac {1}{2}\right )}{1288408}-\frac {95499\,x}{10000}-\frac {-\frac {115810726791\,x^3}{7320500000}-\frac {3950432948061\,x^2}{292820000000}+\frac {131252111833\,x}{146410000000}+\frac {579053717731}{292820000000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}}-\frac {2187\,x^2}{2000} \] Input:

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

Output:

(24279*log(x + 3/5))/503284375 - (25059237*log(x - 1/2))/1288408 - (95499* 
x)/10000 - ((131252111833*x)/146410000000 - (3950432948061*x^2)/2928200000 
00 - (115810726791*x^3)/7320500000 + 579053717731/292820000000)/(x^3/5 - ( 
59*x^2)/100 - (3*x)/50 + x^4 + 9/100) - (2187*x^2)/2000
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.92 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {38846400 \,\mathrm {log}\left (5 x +3\right ) x^{4}+7769280 \,\mathrm {log}\left (5 x +3\right ) x^{3}-22919376 \,\mathrm {log}\left (5 x +3\right ) x^{2}-2330784 \,\mathrm {log}\left (5 x +3\right ) x +3496176 \,\mathrm {log}\left (5 x +3\right )-15662023125000 \,\mathrm {log}\left (2 x -1\right ) x^{4}-3132404625000 \,\mathrm {log}\left (2 x -1\right ) x^{3}[phone] \,\mathrm {log}\left (2 x -1\right ) x^{2}[phone] \,\mathrm {log}\left (2 x -1\right ) x -1409582081250 \,\mathrm {log}\left (2 x -1\right )-880546342500 x^{6}-7866213993000 x^{5}-87664371177900 x^{4}[phone] x^{2}[phone] x -9390526255517}{805255000000 x^{4}[phone] x^{3}-475100450000 x^{2}-48315300000 x [phone]} \] Input:

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

Output:

(38846400*log(5*x + 3)*x**4 + 7769280*log(5*x + 3)*x**3 - 22919376*log(5*x 
 + 3)*x**2 - 2330784*log(5*x + 3)*x + 3496176*log(5*x + 3) - 1566202312500 
0*log(2*x - 1)*x**4 - 3132404625000*log(2*x - 1)*x**3 + 9240593643750*log( 
2*x - 1)*x**2 + 939721387500*log(2*x - 1)*x - 1409582081250*log(2*x - 1) - 
 880546342500*x**6 - 7866213993000*x**5 - 87664371177900*x**4 + 6236691253 
9107*x**2 + 3784756314218*x - 9390526255517)/(8052550000*(100*x**4 + 20*x* 
*3 - 59*x**2 - 6*x + 9))