Integrand size = 22, antiderivative size = 88 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {49}{5 (2+3 x)^5}+\frac {707}{4 (2+3 x)^4}+\frac {6934}{3 (2+3 x)^3}+\frac {28555}{(2+3 x)^2}+\frac {424975}{2+3 x}-\frac {15125}{2 (3+5 x)^2}+\frac {277750}{3+5 x}-2958125 \log (2+3 x)+2958125 \log (3+5 x) \]
49/5/(2+3*x)^5+707/4/(2+3*x)^4+6934/3/(2+3*x)^3+28555/(2+3*x)^2+424975/(2+ 3*x)-15125/2/(3+5*x)^2+277750/(3+5*x)-2958125*ln(2+3*x)+2958125*ln(3+5*x)
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {49}{5 (2+3 x)^5}+\frac {707}{4 (2+3 x)^4}+\frac {6934}{3 (2+3 x)^3}+\frac {28555}{(2+3 x)^2}+\frac {424975}{2+3 x}-\frac {15125}{2 (3+5 x)^2}+\frac {277750}{3+5 x}-2958125 \log (5 (2+3 x))+2958125 \log (3+5 x) \]
49/(5*(2 + 3*x)^5) + 707/(4*(2 + 3*x)^4) + 6934/(3*(2 + 3*x)^3) + 28555/(2 + 3*x)^2 + 424975/(2 + 3*x) - 15125/(2*(3 + 5*x)^2) + 277750/(3 + 5*x) - 2958125*Log[5*(2 + 3*x)] + 2958125*Log[3 + 5*x]
Time = 0.23 (sec) , antiderivative size = 88, 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.
\(\displaystyle \int \frac {(1-2 x)^2}{(3 x+2)^6 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {14790625}{5 x+3}-\frac {1388750}{(5 x+3)^2}+\frac {75625}{(5 x+3)^3}-\frac {8874375}{3 x+2}-\frac {1274925}{(3 x+2)^2}-\frac {171330}{(3 x+2)^3}-\frac {20802}{(3 x+2)^4}-\frac {2121}{(3 x+2)^5}-\frac {147}{(3 x+2)^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {424975}{3 x+2}+\frac {277750}{5 x+3}+\frac {28555}{(3 x+2)^2}-\frac {15125}{2 (5 x+3)^2}+\frac {6934}{3 (3 x+2)^3}+\frac {707}{4 (3 x+2)^4}+\frac {49}{5 (3 x+2)^5}-2958125 \log (3 x+2)+2958125 \log (5 x+3)\) |
49/(5*(2 + 3*x)^5) + 707/(4*(2 + 3*x)^4) + 6934/(3*(2 + 3*x)^3) + 28555/(2 + 3*x)^2 + 424975/(2 + 3*x) - 15125/(2*(3 + 5*x)^2) + 277750/(3 + 5*x) - 2958125*Log[2 + 3*x] + 2958125*Log[3 + 5*x]
3.14.35.3.1 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 = 2.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72
method | result | size |
norman | \(\frac {1198040625 x^{6}+7589365500 x^{4}+\frac {3321676301}{4} x +\frac {9344716875}{2} x^{5}+\frac {9598703854}{3} x^{2}+\frac {26287200325}{4} x^{3}+\frac {897608377}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-2958125 \ln \left (2+3 x \right )+2958125 \ln \left (3+5 x \right )\) | \(63\) |
risch | \(\frac {1198040625 x^{6}+7589365500 x^{4}+\frac {3321676301}{4} x +\frac {9344716875}{2} x^{5}+\frac {9598703854}{3} x^{2}+\frac {26287200325}{4} x^{3}+\frac {897608377}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-2958125 \ln \left (2+3 x \right )+2958125 \ln \left (3+5 x \right )\) | \(64\) |
default | \(\frac {49}{5 \left (2+3 x \right )^{5}}+\frac {707}{4 \left (2+3 x \right )^{4}}+\frac {6934}{3 \left (2+3 x \right )^{3}}+\frac {28555}{\left (2+3 x \right )^{2}}+\frac {424975}{2+3 x}-\frac {15125}{2 \left (3+5 x \right )^{2}}+\frac {277750}{3+5 x}-2958125 \ln \left (2+3 x \right )+2958125 \ln \left (3+5 x \right )\) | \(81\) |
parallelrisch | \(-\frac {408931199520 x -123360912000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+317944008000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-26580528000000 \ln \left (x +\frac {3}{5}\right ) x +123360912000000 \ln \left (\frac {2}{3}+x \right ) x^{2}+26580528000000 \ln \left (\frac {2}{3}+x \right ) x +34553986960599 x^{5}+21269777702580 x^{6}+5452970890275 x^{7}+14571960395640 x^{3}+29925654629130 x^{4}+3782613599120 x^{2}+491484186000000 \ln \left (\frac {2}{3}+x \right ) x^{4}+2453587200000 \ln \left (\frac {2}{3}+x \right )+51755355000000 \ln \left (\frac {2}{3}+x \right ) x^{7}-51755355000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-2453587200000 \ln \left (x +\frac {3}{5}\right )+455677147800000 \ln \left (\frac {2}{3}+x \right ) x^{5}-317944008000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-455677147800000 \ln \left (x +\frac {3}{5}\right ) x^{5}-491484186000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+234624276000000 \ln \left (\frac {2}{3}+x \right ) x^{6}-234624276000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{2880 \left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}\) | \(185\) |
(1198040625*x^6+7589365500*x^4+3321676301/4*x+9344716875/2*x^5+9598703854/ 3*x^2+26287200325/4*x^3+897608377/10)/(2+3*x)^5/(3+5*x)^2-2958125*ln(2+3*x )+2958125*ln(3+5*x)
Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.76 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {71882437500 \, x^{6} + 280341506250 \, x^{5} + 455361930000 \, x^{4} + 394308004875 \, x^{3} + 191974077080 \, x^{2} + 177487500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (5 \, x + 3\right ) - 177487500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (3 \, x + 2\right ) + 49825144515 \, x + 5385650262}{60 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \]
1/60*(71882437500*x^6 + 280341506250*x^5 + 455361930000*x^4 + 394308004875 *x^3 + 191974077080*x^2 + 177487500*(6075*x^7 + 27540*x^6 + 53487*x^5 + 57 690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)*log(5*x + 3) - 177487500*( 6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 312 0*x + 288)*log(3*x + 2) + 49825144515*x + 5385650262)/(6075*x^7 + 27540*x^ 6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {71882437500 x^{6} + 280341506250 x^{5} + 455361930000 x^{4} + 394308004875 x^{3} + 191974077080 x^{2} + 49825144515 x + 5385650262}{364500 x^{7} + 1652400 x^{6} + 3209220 x^{5} + 3461400 x^{4} + 2239200 x^{3} + 868800 x^{2} + 187200 x + 17280} + 2958125 \log {\left (x + \frac {3}{5} \right )} - 2958125 \log {\left (x + \frac {2}{3} \right )} \]
(71882437500*x**6 + 280341506250*x**5 + 455361930000*x**4 + 394308004875*x **3 + 191974077080*x**2 + 49825144515*x + 5385650262)/(364500*x**7 + 16524 00*x**6 + 3209220*x**5 + 3461400*x**4 + 2239200*x**3 + 868800*x**2 + 18720 0*x + 17280) + 2958125*log(x + 3/5) - 2958125*log(x + 2/3)
Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {71882437500 \, x^{6} + 280341506250 \, x^{5} + 455361930000 \, x^{4} + 394308004875 \, x^{3} + 191974077080 \, x^{2} + 49825144515 \, x + 5385650262}{60 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} + 2958125 \, \log \left (5 \, x + 3\right ) - 2958125 \, \log \left (3 \, x + 2\right ) \]
1/60*(71882437500*x^6 + 280341506250*x^5 + 455361930000*x^4 + 394308004875 *x^3 + 191974077080*x^2 + 49825144515*x + 5385650262)/(6075*x^7 + 27540*x^ 6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288) + 295812 5*log(5*x + 3) - 2958125*log(3*x + 2)
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {71882437500 \, x^{6} + 280341506250 \, x^{5} + 455361930000 \, x^{4} + 394308004875 \, x^{3} + 191974077080 \, x^{2} + 49825144515 \, x + 5385650262}{60 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{5}} + 2958125 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 2958125 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]
1/60*(71882437500*x^6 + 280341506250*x^5 + 455361930000*x^4 + 394308004875 *x^3 + 191974077080*x^2 + 49825144515*x + 5385650262)/((5*x + 3)^2*(3*x + 2)^5) + 2958125*log(abs(5*x + 3)) - 2958125*log(abs(3*x + 2))
Time = 1.37 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {\frac {591625\,x^6}{3}+\frac {1538225\,x^5}{2}+\frac {101191540\,x^4}{81}+\frac {1051488013\,x^3}{972}+\frac {9598703854\,x^2}{18225}+\frac {3321676301\,x}{24300}+\frac {897608377}{60750}}{x^7+\frac {68\,x^6}{15}+\frac {1981\,x^5}{225}+\frac {1282\,x^4}{135}+\frac {2488\,x^3}{405}+\frac {2896\,x^2}{1215}+\frac {208\,x}{405}+\frac {32}{675}}-5916250\,\mathrm {atanh}\left (30\,x+19\right ) \]