Integrand size = 22, antiderivative size = 75 \[ \int \frac {1}{(1-2 x) (2+3 x)^3 (3+5 x)^3} \, dx=\frac {27}{14 (2+3 x)^2}+\frac {2889}{49 (2+3 x)}-\frac {125}{22 (3+5 x)^2}+\frac {12125}{121 (3+5 x)}-\frac {32 \log (1-2 x)}{456533}-\frac {204228}{343} \log (2+3 x)+\frac {792500 \log (3+5 x)}{1331} \] Output:
27/14/(2+3*x)^2+2889/(98+147*x)-125/22/(3+5*x)^2+12125/(363+605*x)-32/4565 33*ln(1-2*x)-204228/343*ln(2+3*x)+792500/1331*ln(3+5*x)
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-2 x) (2+3 x)^3 (3+5 x)^3} \, dx=\frac {27}{14 (2+3 x)^2}-\frac {125}{22 (3+5 x)^2}+\frac {2889}{98+147 x}+\frac {12125}{363+605 x}-\frac {32 \log (1-2 x)}{456533}-\frac {204228}{343} \log (4+6 x)+\frac {792500 \log (6+10 x)}{1331} \] Input:
Integrate[1/((1 - 2*x)*(2 + 3*x)^3*(3 + 5*x)^3),x]
Output:
27/(14*(2 + 3*x)^2) - 125/(22*(3 + 5*x)^2) + 2889/(98 + 147*x) + 12125/(36 3 + 605*x) - (32*Log[1 - 2*x])/456533 - (204228*Log[4 + 6*x])/343 + (79250 0*Log[6 + 10*x])/1331
Time = 0.24 (sec) , antiderivative size = 75, 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.
\(\displaystyle \int \frac {1}{(1-2 x) (3 x+2)^3 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {612684}{343 (3 x+2)}+\frac {3962500}{1331 (5 x+3)}-\frac {8667}{49 (3 x+2)^2}-\frac {60625}{121 (5 x+3)^2}-\frac {81}{7 (3 x+2)^3}+\frac {625}{11 (5 x+3)^3}-\frac {64}{456533 (2 x-1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2889}{49 (3 x+2)}+\frac {12125}{121 (5 x+3)}+\frac {27}{14 (3 x+2)^2}-\frac {125}{22 (5 x+3)^2}-\frac {32 \log (1-2 x)}{456533}-\frac {204228}{343} \log (3 x+2)+\frac {792500 \log (5 x+3)}{1331}\) |
Input:
Int[1/((1 - 2*x)*(2 + 3*x)^3*(3 + 5*x)^3),x]
Output:
27/(14*(2 + 3*x)^2) + 2889/(49*(2 + 3*x)) - 125/(22*(3 + 5*x)^2) + 12125/( 121*(3 + 5*x)) - (32*Log[1 - 2*x])/456533 - (204228*Log[2 + 3*x])/343 + (7 92500*Log[3 + 5*x])/1331
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]))
Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.75
method | result | size |
norman | \(\frac {\frac {2053290}{121} x^{2}+\frac {52953300}{5929} x^{3}+\frac {63622288}{5929} x +\frac {26779805}{11858}}{\left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {32 \ln \left (-1+2 x \right )}{456533}-\frac {204228 \ln \left (2+3 x \right )}{343}+\frac {792500 \ln \left (3+5 x \right )}{1331}\) | \(56\) |
risch | \(\frac {\frac {2053290}{121} x^{2}+\frac {52953300}{5929} x^{3}+\frac {63622288}{5929} x +\frac {26779805}{11858}}{\left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {32 \ln \left (-1+2 x \right )}{456533}-\frac {204228 \ln \left (2+3 x \right )}{343}+\frac {792500 \ln \left (3+5 x \right )}{1331}\) | \(57\) |
default | \(-\frac {125}{22 \left (3+5 x \right )^{2}}+\frac {12125}{121 \left (3+5 x \right )}+\frac {792500 \ln \left (3+5 x \right )}{1331}+\frac {27}{14 \left (2+3 x \right )^{2}}+\frac {2889}{49 \left (2+3 x \right )}-\frac {204228 \ln \left (2+3 x \right )}{343}-\frac {32 \ln \left (-1+2 x \right )}{456533}\) | \(62\) |
parallelrisch | \(-\frac {4403604981600 \ln \left (\frac {2}{3}+x \right ) x^{4}+518400 \ln \left (x -\frac {1}{2}\right ) x^{4}-4403605500000 \ln \left (x +\frac {3}{5}\right ) x^{4}+11155799286720 \ln \left (\frac {2}{3}+x \right ) x^{3}+1313280 \ln \left (x -\frac {1}{2}\right ) x^{3}-11155800600000 \ln \left (x +\frac {3}{5}\right ) x^{3}+463960121625 x^{4}+10588223533536 \ln \left (\frac {2}{3}+x \right ) x^{2}+1246464 \ln \left (x -\frac {1}{2}\right ) x^{2}-10588224780000 \ln \left (x +\frac {3}{5}\right ) x^{2}+881792546250 x^{3}+4462319714688 \ln \left (\frac {2}{3}+x \right ) x +525312 \ln \left (x -\frac {1}{2}\right ) x -4462320240000 \ln \left (x +\frac {3}{5}\right ) x +557777788645 x^{2}+704576797056 \ln \left (\frac {2}{3}+x \right )+82944 \ln \left (x -\frac {1}{2}\right )-704576880000 \ln \left (x +\frac {3}{5}\right )+117424291908 x}{32870376 \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(156\) |
Input:
int(1/(1-2*x)/(2+3*x)^3/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
(2053290/121*x^2+52953300/5929*x^3+63622288/5929*x+26779805/11858)/(2+3*x) ^2/(3+5*x)^2-32/456533*ln(-1+2*x)-204228/343*ln(2+3*x)+792500/1331*ln(3+5* x)
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (61) = 122\).
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.64 \[ \int \frac {1}{(1-2 x) (2+3 x)^3 (3+5 x)^3} \, dx=\frac {8154808200 \, x^{3} + 15494126340 \, x^{2} + 543655000 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (5 \, x + 3\right ) - 543654936 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (3 \, x + 2\right ) - 64 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (2 \, x - 1\right ) + 9797832352 \, x + 2062044985}{913066 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \] Input:
integrate(1/(1-2*x)/(2+3*x)^3/(3+5*x)^3,x, algorithm="fricas")
Output:
1/913066*(8154808200*x^3 + 15494126340*x^2 + 543655000*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(5*x + 3) - 543654936*(225*x^4 + 570*x^3 + 541* x^2 + 228*x + 36)*log(3*x + 2) - 64*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(2*x - 1) + 9797832352*x + 2062044985)/(225*x^4 + 570*x^3 + 541*x^ 2 + 228*x + 36)
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x) (2+3 x)^3 (3+5 x)^3} \, dx=- \frac {- 105906600 x^{3} - 201222420 x^{2} - 127244576 x - 26779805}{2668050 x^{4} + 6759060 x^{3} + 6415178 x^{2} + 2703624 x + 426888} - \frac {32 \log {\left (x - \frac {1}{2} \right )}}{456533} + \frac {792500 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {204228 \log {\left (x + \frac {2}{3} \right )}}{343} \] Input:
integrate(1/(1-2*x)/(2+3*x)**3/(3+5*x)**3,x)
Output:
-(-105906600*x**3 - 201222420*x**2 - 127244576*x - 26779805)/(2668050*x**4 + 6759060*x**3 + 6415178*x**2 + 2703624*x + 426888) - 32*log(x - 1/2)/456 533 + 792500*log(x + 3/5)/1331 - 204228*log(x + 2/3)/343
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(1-2 x) (2+3 x)^3 (3+5 x)^3} \, dx=\frac {105906600 \, x^{3} + 201222420 \, x^{2} + 127244576 \, x + 26779805}{11858 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} + \frac {792500}{1331} \, \log \left (5 \, x + 3\right ) - \frac {204228}{343} \, \log \left (3 \, x + 2\right ) - \frac {32}{456533} \, \log \left (2 \, x - 1\right ) \] Input:
integrate(1/(1-2*x)/(2+3*x)^3/(3+5*x)^3,x, algorithm="maxima")
Output:
1/11858*(105906600*x^3 + 201222420*x^2 + 127244576*x + 26779805)/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36) + 792500/1331*log(5*x + 3) - 204228/343* log(3*x + 2) - 32/456533*log(2*x - 1)
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x) (2+3 x)^3 (3+5 x)^3} \, dx=\frac {105906600 \, x^{3} + 201222420 \, x^{2} + 127244576 \, x + 26779805}{11858 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{2}} + \frac {792500}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {204228}{343} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {32}{456533} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \] Input:
integrate(1/(1-2*x)/(2+3*x)^3/(3+5*x)^3,x, algorithm="giac")
Output:
1/11858*(105906600*x^3 + 201222420*x^2 + 127244576*x + 26779805)/((5*x + 3 )^2*(3*x + 2)^2) + 792500/1331*log(abs(5*x + 3)) - 204228/343*log(abs(3*x + 2)) - 32/456533*log(abs(2*x - 1))
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1-2 x) (2+3 x)^3 (3+5 x)^3} \, dx=\frac {792500\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {204228\,\ln \left (x+\frac {2}{3}\right )}{343}-\frac {32\,\ln \left (x-\frac {1}{2}\right )}{456533}+\frac {\frac {235348\,x^3}{5929}+\frac {136886\,x^2}{1815}+\frac {63622288\,x}{1334025}+\frac {5355961}{533610}}{x^4+\frac {38\,x^3}{15}+\frac {541\,x^2}{225}+\frac {76\,x}{75}+\frac {4}{25}} \] Input:
int(-1/((2*x - 1)*(3*x + 2)^3*(5*x + 3)^3),x)
Output:
(792500*log(x + 3/5))/1331 - (204228*log(x + 2/3))/343 - (32*log(x - 1/2)) /456533 + ((63622288*x)/1334025 + (136886*x^2)/1815 + (235348*x^3)/5929 + 5355961/533610)/((76*x)/75 + (541*x^2)/225 + (38*x^3)/15 + x^4 + 4/25)
Time = 0.16 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.51 \[ \int \frac {1}{(1-2 x) (2+3 x)^3 (3+5 x)^3} \, dx=\frac {2324125125000 \,\mathrm {log}\left (5 x +3\right ) x^{4}+5887783650000 \,\mathrm {log}\left (5 x +3\right ) x^{3}+5588229745000 \,\mathrm {log}\left (5 x +3\right ) x^{2}+2355113460000 \,\mathrm {log}\left (5 x +3\right ) x +371860020000 \,\mathrm {log}\left (5 x +3\right )-2324124851400 \,\mathrm {log}\left (3 x +2\right ) x^{4}-5887782956880 \,\mathrm {log}\left (3 x +2\right ) x^{3}-5588229087144 \,\mathrm {log}\left (3 x +2\right ) x^{2}-2355113182752 \,\mathrm {log}\left (3 x +2\right ) x -371859976224 \,\mathrm {log}\left (3 x +2\right )-273600 \,\mathrm {log}\left (2 x -1\right ) x^{4}-693120 \,\mathrm {log}\left (2 x -1\right ) x^{3}-657856 \,\mathrm {log}\left (2 x -1\right ) x^{2}-277248 \,\mathrm {log}\left (2 x -1\right ) x -43776 \,\mathrm {log}\left (2 x -1\right )-61161061500 x^{4}+147330025920 x^{2}+124182272368 x +29393084875}{3903357150 x^{4}+9888504780 x^{3}+9385405414 x^{2}+3955401912 x +624537144} \] Input:
int(1/(1-2*x)/(2+3*x)^3/(3+5*x)^3,x)
Output:
(2324125125000*log(5*x + 3)*x**4 + 5887783650000*log(5*x + 3)*x**3 + 55882 29745000*log(5*x + 3)*x**2 + 2355113460000*log(5*x + 3)*x + 371860020000*l og(5*x + 3) - 2324124851400*log(3*x + 2)*x**4 - 5887782956880*log(3*x + 2) *x**3 - 5588229087144*log(3*x + 2)*x**2 - 2355113182752*log(3*x + 2)*x - 3 71859976224*log(3*x + 2) - 273600*log(2*x - 1)*x**4 - 693120*log(2*x - 1)* x**3 - 657856*log(2*x - 1)*x**2 - 277248*log(2*x - 1)*x - 43776*log(2*x - 1) - 61161061500*x**4 + 147330025920*x**2 + 124182272368*x + 29393084875)/ (17348254*(225*x**4 + 570*x**3 + 541*x**2 + 228*x + 36))