Integrand size = 22, antiderivative size = 59 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {16807}{3872 (1-2 x)^2}-\frac {228095}{21296 (1-2 x)}-\frac {243 x}{200}-\frac {1}{166375 (3+5 x)}-\frac {1034145 \log (1-2 x)}{234256}+\frac {171 \log (3+5 x)}{1830125} \]
16807/3872/(1-2*x)^2-228095/21296/(1-2*x)-243/200*x-1/166375/(3+5*x)-10341 45/234256*ln(1-2*x)+171/1830125*ln(3+5*x)
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.93 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {\frac {254205875}{(1-2 x)^2}+35577630 (1-2 x)+\frac {627261250}{-1+2 x}-\frac {352}{3+5 x}-258536250 \log (1-2 x)+5472 \log (6+10 x)}{58564000} \]
(254205875/(1 - 2*x)^2 + 35577630*(1 - 2*x) + 627261250/(-1 + 2*x) - 352/( 3 + 5*x) - 258536250*Log[1 - 2*x] + 5472*Log[6 + 10*x])/58564000
Time = 0.19 (sec) , antiderivative size = 59, 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.
\(\displaystyle \int \frac {(3 x+2)^5}{(1-2 x)^3 (5 x+3)^2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {171}{366025 (5 x+3)}+\frac {1}{33275 (5 x+3)^2}-\frac {1034145}{117128 (2 x-1)}-\frac {228095}{10648 (2 x-1)^2}-\frac {16807}{968 (2 x-1)^3}-\frac {243}{200}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {243 x}{200}-\frac {228095}{21296 (1-2 x)}-\frac {1}{166375 (5 x+3)}+\frac {16807}{3872 (1-2 x)^2}-\frac {1034145 \log (1-2 x)}{234256}+\frac {171 \log (5 x+3)}{1830125}\) |
16807/(3872*(1 - 2*x)^2) - 228095/(21296*(1 - 2*x)) - (243*x)/200 - 1/(166 375*(3 + 5*x)) - (1034145*Log[1 - 2*x])/234256 + (171*Log[3 + 5*x])/183012 5
3.17.87.3.1 Defintions of rubi rules used
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 = 0.89 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {243 x}{200}+\frac {\frac {142559343}{1331000} x^{2}+\frac {172572003}{5324000} x -\frac {101742407}{5324000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {1034145 \ln \left (-1+2 x \right )}{234256}+\frac {171 \ln \left (3+5 x \right )}{1830125}\) | \(47\) |
default | \(-\frac {243 x}{200}-\frac {1}{166375 \left (3+5 x \right )}+\frac {171 \ln \left (3+5 x \right )}{1830125}+\frac {16807}{3872 \left (-1+2 x \right )^{2}}+\frac {228095}{21296 \left (-1+2 x \right )}-\frac {1034145 \ln \left (-1+2 x \right )}{234256}\) | \(48\) |
norman | \(\frac {\frac {6453842}{99825} x^{2}-\frac {12634939}{798600} x +\frac {109504799}{798600} x^{3}-\frac {243}{10} x^{4}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {1034145 \ln \left (-1+2 x \right )}{234256}+\frac {171 \ln \left (3+5 x \right )}{1830125}\) | \(52\) |
parallelrisch | \(\frac {164160 \ln \left (x +\frac {3}{5}\right ) x^{3}-7756087500 \ln \left (x -\frac {1}{2}\right ) x^{3}-2134657800 x^{4}-65664 \ln \left (x +\frac {3}{5}\right ) x^{2}+3102435000 \ln \left (x -\frac {1}{2}\right ) x^{2}+12045527890 x^{3}-57456 \ln \left (x +\frac {3}{5}\right ) x +2714630625 \ln \left (x -\frac {1}{2}\right ) x +5679380960 x^{2}+24624 \ln \left (x +\frac {3}{5}\right )-1163413125 \ln \left (x -\frac {1}{2}\right )-1389843290 x}{87846000 \left (-1+2 x \right )^{2} \left (3+5 x \right )}\) | \(98\) |
-243/200*x+20*(142559343/26620000*x^2+172572003/106480000*x-101742407/1064 80000)/(-1+2*x)^2/(3+5*x)-1034145/234256*ln(-1+2*x)+171/1830125*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.44 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {1423105200 \, x^{4} - 569242080 \, x^{3} - 6770697912 \, x^{2} - 5472 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 258536250 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 1684826253 \, x + 1119166477}{58564000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
-1/58564000*(1423105200*x^4 - 569242080*x^3 - 6770697912*x^2 - 5472*(20*x^ 3 - 8*x^2 - 7*x + 3)*log(5*x + 3) + 258536250*(20*x^3 - 8*x^2 - 7*x + 3)*l og(2*x - 1) - 1684826253*x + 1119166477)/(20*x^3 - 8*x^2 - 7*x + 3)
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)^2} \, dx=- \frac {243 x}{200} - \frac {- 570237372 x^{2} - 172572003 x + 101742407}{106480000 x^{3} - 42592000 x^{2} - 37268000 x + 15972000} - \frac {1034145 \log {\left (x - \frac {1}{2} \right )}}{234256} + \frac {171 \log {\left (x + \frac {3}{5} \right )}}{1830125} \]
-243*x/200 - (-570237372*x**2 - 172572003*x + 101742407)/(106480000*x**3 - 42592000*x**2 - 37268000*x + 15972000) - 1034145*log(x - 1/2)/234256 + 17 1*log(x + 3/5)/1830125
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)^2} \, dx=-\frac {243}{200} \, x + \frac {570237372 \, x^{2} + 172572003 \, x - 101742407}{5324000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac {171}{1830125} \, \log \left (5 \, x + 3\right ) - \frac {1034145}{234256} \, \log \left (2 \, x - 1\right ) \]
-243/200*x + 1/5324000*(570237372*x^2 + 172572003*x - 101742407)/(20*x^3 - 8*x^2 - 7*x + 3) + 171/1830125*log(5*x + 3) - 1034145/234256*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.41 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {{\left (5 \, x + 3\right )} {\left (\frac {389138447}{5 \, x + 3} - \frac {1420901823}{{\left (5 \, x + 3\right )}^{2}} - 14231052\right )}}{14641000 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2}} - \frac {1}{166375 \, {\left (5 \, x + 3\right )}} + \frac {8829}{2000} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {1034145}{234256} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]
1/14641000*(5*x + 3)*(389138447/(5*x + 3) - 1420901823/(5*x + 3)^2 - 14231 052)/(11/(5*x + 3) - 2)^2 - 1/166375/(5*x + 3) + 8829/2000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 1034145/234256*log(abs(-11/(5*x + 3) + 2))
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^5}{(1-2 x)^3 (3+5 x)^2} \, dx=\frac {171\,\ln \left (x+\frac {3}{5}\right )}{1830125}-\frac {1034145\,\ln \left (x-\frac {1}{2}\right )}{234256}-\frac {243\,x}{200}-\frac {\frac {142559343\,x^2}{26620000}+\frac {172572003\,x}{106480000}-\frac {101742407}{106480000}}{-x^3+\frac {2\,x^2}{5}+\frac {7\,x}{20}-\frac {3}{20}} \]