Integrand size = 22, antiderivative size = 69 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)} \, dx=\frac {823543}{2816 (1-2 x)^2}-\frac {5764801}{3872 (1-2 x)}-\frac {26161299 x}{20000}-\frac {792423 x^2}{2000}-\frac {40581 x^3}{400}-\frac {2187 x^4}{160}-\frac {269063263 \log (1-2 x)}{170368}+\frac {\log (3+5 x)}{4159375} \] Output:
823543/2816/(1-2*x)^2-5764801/(3872-7744*x)-26161299/20000*x-792423/2000*x ^2-40581/400*x^3-2187/160*x^4-269063263/170368*ln(1-2*x)+1/4159375*ln(3+5* x)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)} \, dx=\frac {-\frac {55 \left (35985148011-83615877112 x-42333890544 x^2+72578051568 x^3+23090763960 x^4+6797973600 x^5+1058508000 x^6\right )}{(1-2 x)^2}-1681645393750 \log (5-10 x)+256 \log (3+5 x)}{1064800000} \] Input:
Integrate[(2 + 3*x)^7/((1 - 2*x)^3*(3 + 5*x)),x]
Output:
((-55*(35985148011 - 83615877112*x - 42333890544*x^2 + 72578051568*x^3 + 2 3090763960*x^4 + 6797973600*x^5 + 1058508000*x^6))/(1 - 2*x)^2 - 168164539 3750*Log[5 - 10*x] + 256*Log[3 + 5*x])/1064800000
Time = 0.23 (sec) , antiderivative size = 69, 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.
\(\displaystyle \int \frac {(3 x+2)^7}{(1-2 x)^3 (5 x+3)} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {2187 x^3}{40}-\frac {121743 x^2}{400}-\frac {792423 x}{1000}-\frac {269063263}{85184 (2 x-1)}+\frac {1}{831875 (5 x+3)}-\frac {5764801}{1936 (2 x-1)^2}-\frac {823543}{704 (2 x-1)^3}-\frac {26161299}{20000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2187 x^4}{160}-\frac {40581 x^3}{400}-\frac {792423 x^2}{2000}-\frac {26161299 x}{20000}-\frac {5764801}{3872 (1-2 x)}+\frac {823543}{2816 (1-2 x)^2}-\frac {269063263 \log (1-2 x)}{170368}+\frac {\log (5 x+3)}{4159375}\) |
Input:
Int[(2 + 3*x)^7/((1 - 2*x)^3*(3 + 5*x)),x]
Output:
823543/(2816*(1 - 2*x)^2) - 5764801/(3872*(1 - 2*x)) - (26161299*x)/20000 - (792423*x^2)/2000 - (40581*x^3)/400 - (2187*x^4)/160 - (269063263*Log[1 - 2*x])/170368 + Log[3 + 5*x]/4159375
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 = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {2187 x^{4}}{160}-\frac {40581 x^{3}}{400}-\frac {792423 x^{2}}{2000}-\frac {26161299 x}{20000}+\frac {\frac {5764801 x}{1936}-\frac {37059435}{30976}}{\left (-1+2 x \right )^{2}}-\frac {269063263 \ln \left (-1+2 x \right )}{170368}+\frac {\ln \left (3+5 x \right )}{4159375}\) | \(50\) |
default | \(-\frac {2187 x^{4}}{160}-\frac {40581 x^{3}}{400}-\frac {792423 x^{2}}{2000}-\frac {26161299 x}{20000}+\frac {\ln \left (3+5 x \right )}{4159375}+\frac {823543}{2816 \left (-1+2 x \right )^{2}}+\frac {5764801}{3872 \left (-1+2 x \right )}-\frac {269063263 \ln \left (-1+2 x \right )}{170368}\) | \(54\) |
norman | \(\frac {-\frac {15081178733}{4840000} x +\frac {46568620647}{4840000} x^{2}-\frac {37488663}{10000} x^{3}-\frac {4770819}{4000} x^{4}-\frac {70227}{200} x^{5}-\frac {2187}{40} x^{6}}{\left (-1+2 x \right )^{2}}-\frac {269063263 \ln \left (-1+2 x \right )}{170368}+\frac {\ln \left (3+5 x \right )}{4159375}\) | \(55\) |
parallelrisch | \(-\frac {29108970000 x^{6}+186944274000 x^{5}+634996008900 x^{4}+3363290787500 \ln \left (x -\frac {1}{2}\right ) x^{2}-512 \ln \left (x +\frac {3}{5}\right ) x^{2}+1995896418120 x^{3}-3363290787500 \ln \left (x -\frac {1}{2}\right ) x +512 \ln \left (x +\frac {3}{5}\right ) x -5122548271170 x^{2}+840822696875 \ln \left (x -\frac {1}{2}\right )-128 \ln \left (x +\frac {3}{5}\right )+1658929660630 x}{532400000 \left (-1+2 x \right )^{2}}\) | \(83\) |
Input:
int((2+3*x)^7/(1-2*x)^3/(3+5*x),x,method=_RETURNVERBOSE)
Output:
-2187/160*x^4-40581/400*x^3-792423/2000*x^2-26161299/20000*x+4*(5764801/77 44*x-37059435/123904)/(-1+2*x)^2-269063263/170368*ln(-1+2*x)+1/4159375*ln( 3+5*x)
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.16 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)} \, dx=-\frac {58217940000 \, x^{6} + 373888548000 \, x^{5} + 1269992017800 \, x^{4} + 3991792836240 \, x^{3} - 5149424229840 \, x^{2} - 256 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 1681645393750 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 1777812991240 \, x + 1273918078125}{1064800000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:
integrate((2+3*x)^7/(1-2*x)^3/(3+5*x),x, algorithm="fricas")
Output:
-1/1064800000*(58217940000*x^6 + 373888548000*x^5 + 1269992017800*x^4 + 39 91792836240*x^3 - 5149424229840*x^2 - 256*(4*x^2 - 4*x + 1)*log(5*x + 3) + 1681645393750*(4*x^2 - 4*x + 1)*log(2*x - 1) - 1777812991240*x + 12739180 78125)/(4*x^2 - 4*x + 1)
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)} \, dx=- \frac {2187 x^{4}}{160} - \frac {40581 x^{3}}{400} - \frac {792423 x^{2}}{2000} - \frac {26161299 x}{20000} - \frac {37059435 - 92236816 x}{123904 x^{2} - 123904 x + 30976} - \frac {269063263 \log {\left (x - \frac {1}{2} \right )}}{170368} + \frac {\log {\left (x + \frac {3}{5} \right )}}{4159375} \] Input:
integrate((2+3*x)**7/(1-2*x)**3/(3+5*x),x)
Output:
-2187*x**4/160 - 40581*x**3/400 - 792423*x**2/2000 - 26161299*x/20000 - (3 7059435 - 92236816*x)/(123904*x**2 - 123904*x + 30976) - 269063263*log(x - 1/2)/170368 + log(x + 3/5)/4159375
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)} \, dx=-\frac {2187}{160} \, x^{4} - \frac {40581}{400} \, x^{3} - \frac {792423}{2000} \, x^{2} - \frac {26161299}{20000} \, x + \frac {823543 \, {\left (112 \, x - 45\right )}}{30976 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1}{4159375} \, \log \left (5 \, x + 3\right ) - \frac {269063263}{170368} \, \log \left (2 \, x - 1\right ) \] Input:
integrate((2+3*x)^7/(1-2*x)^3/(3+5*x),x, algorithm="maxima")
Output:
-2187/160*x^4 - 40581/400*x^3 - 792423/2000*x^2 - 26161299/20000*x + 82354 3/30976*(112*x - 45)/(4*x^2 - 4*x + 1) + 1/4159375*log(5*x + 3) - 26906326 3/170368*log(2*x - 1)
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)} \, dx=-\frac {2187}{160} \, x^{4} - \frac {40581}{400} \, x^{3} - \frac {792423}{2000} \, x^{2} - \frac {26161299}{20000} \, x + \frac {823543 \, {\left (112 \, x - 45\right )}}{30976 \, {\left (2 \, x - 1\right )}^{2}} + \frac {1}{4159375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {269063263}{170368} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \] Input:
integrate((2+3*x)^7/(1-2*x)^3/(3+5*x),x, algorithm="giac")
Output:
-2187/160*x^4 - 40581/400*x^3 - 792423/2000*x^2 - 26161299/20000*x + 82354 3/30976*(112*x - 45)/(2*x - 1)^2 + 1/4159375*log(abs(5*x + 3)) - 269063263 /170368*log(abs(2*x - 1))
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)} \, dx=\frac {\ln \left (x+\frac {3}{5}\right )}{4159375}-\frac {269063263\,\ln \left (x-\frac {1}{2}\right )}{170368}-\frac {26161299\,x}{20000}+\frac {\frac {5764801\,x}{7744}-\frac {37059435}{123904}}{x^2-x+\frac {1}{4}}-\frac {792423\,x^2}{2000}-\frac {40581\,x^3}{400}-\frac {2187\,x^4}{160} \] Input:
int(-(3*x + 2)^7/((2*x - 1)^3*(5*x + 3)),x)
Output:
log(x + 3/5)/4159375 - (269063263*log(x - 1/2))/170368 - (26161299*x)/2000 0 + ((5764801*x)/7744 - 37059435/123904)/(x^2 - x + 1/4) - (792423*x^2)/20 00 - (40581*x^3)/400 - (2187*x^4)/160
Time = 0.16 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.39 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)} \, dx=\frac {1024 \,\mathrm {log}\left (5 x +3\right ) x^{2}-1024 \,\mathrm {log}\left (5 x +3\right ) x +256 \,\mathrm {log}\left (5 x +3\right )-6726581575000 \,\mathrm {log}\left (2 x -1\right ) x^{2}+6726581575000 \,\mathrm {log}\left (2 x -1\right ) x -1681645393750 \,\mathrm {log}\left (2 x -1\right )-58217940000 x^{6}-373888548000 x^{5}-1269992017800 x^{4}-3991792836240 x^{3}+6927237221080 x^{2}-829464830315}{4259200000 x^{2}-4259200000 x +1064800000} \] Input:
int((2+3*x)^7/(1-2*x)^3/(3+5*x),x)
Output:
(1024*log(5*x + 3)*x**2 - 1024*log(5*x + 3)*x + 256*log(5*x + 3) - 6726581 575000*log(2*x - 1)*x**2 + 6726581575000*log(2*x - 1)*x - 1681645393750*lo g(2*x - 1) - 58217940000*x**6 - 373888548000*x**5 - 1269992017800*x**4 - 3 991792836240*x**3 + 6927237221080*x**2 - 829464830315)/(1064800000*(4*x**2 - 4*x + 1))