Integrand size = 22, antiderivative size = 97 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {7}{3 (2+3 x)^7}-\frac {77}{3 (2+3 x)^6}-\frac {1133}{5 (2+3 x)^5}-\frac {1870}{(2+3 x)^4}-\frac {46475}{3 (2+3 x)^3}-\frac {138875}{(2+3 x)^2}-\frac {1615625}{2+3 x}-\frac {378125}{3+5 x}+9212500 \log (2+3 x)-9212500 \log (3+5 x) \]
-7/3/(2+3*x)^7-77/3/(2+3*x)^6-1133/5/(2+3*x)^5-1870/(2+3*x)^4-46475/3/(2+3 *x)^3-138875/(2+3*x)^2-1615625/(2+3*x)-378125/(3+5*x)+9212500*ln(2+3*x)-92 12500*ln(3+5*x)
Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {7}{3 (2+3 x)^7}-\frac {77}{3 (2+3 x)^6}-\frac {1133}{5 (2+3 x)^5}-\frac {1870}{(2+3 x)^4}-\frac {46475}{3 (2+3 x)^3}-\frac {138875}{(2+3 x)^2}-\frac {1615625}{2+3 x}-\frac {378125}{3+5 x}+9212500 \log (5 (2+3 x))-9212500 \log (3+5 x) \]
-7/(3*(2 + 3*x)^7) - 77/(3*(2 + 3*x)^6) - 1133/(5*(2 + 3*x)^5) - 1870/(2 + 3*x)^4 - 46475/(3*(2 + 3*x)^3) - 138875/(2 + 3*x)^2 - 1615625/(2 + 3*x) - 378125/(3 + 5*x) + 9212500*Log[5*(2 + 3*x)] - 9212500*Log[3 + 5*x]
Time = 0.23 (sec) , antiderivative size = 97, 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.
\(\displaystyle \int \frac {(1-2 x)^2}{(3 x+2)^8 (5 x+3)^2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {46062500}{5 x+3}+\frac {1890625}{(5 x+3)^2}+\frac {27637500}{3 x+2}+\frac {4846875}{(3 x+2)^2}+\frac {833250}{(3 x+2)^3}+\frac {139425}{(3 x+2)^4}+\frac {22440}{(3 x+2)^5}+\frac {3399}{(3 x+2)^6}+\frac {462}{(3 x+2)^7}+\frac {49}{(3 x+2)^8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1615625}{3 x+2}-\frac {378125}{5 x+3}-\frac {138875}{(3 x+2)^2}-\frac {46475}{3 (3 x+2)^3}-\frac {1870}{(3 x+2)^4}-\frac {1133}{5 (3 x+2)^5}-\frac {77}{3 (3 x+2)^6}-\frac {7}{3 (3 x+2)^7}+9212500 \log (3 x+2)-9212500 \log (5 x+3)\) |
-7/(3*(2 + 3*x)^7) - 77/(3*(2 + 3*x)^6) - 1133/(5*(2 + 3*x)^5) - 1870/(2 + 3*x)^4 - 46475/(3*(2 + 3*x)^3) - 138875/(2 + 3*x)^2 - 1615625/(2 + 3*x) - 378125/(3 + 5*x) + 9212500*Log[2 + 3*x] - 9212500*Log[3 + 5*x]
3.14.20.3.1 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 = 2.36 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70
method | result | size |
norman | \(\frac {-68137326675 x^{4}-61781420250 x^{5}-45081841167 x^{3}-31117061250 x^{6}-6715912500 x^{7}-\frac {89469521196}{5} x^{2}-\frac {59178013234}{15} x -\frac {1863616801}{5}}{\left (2+3 x \right )^{7} \left (3+5 x \right )}+9212500 \ln \left (2+3 x \right )-9212500 \ln \left (3+5 x \right )\) | \(68\) |
risch | \(\frac {-68137326675 x^{4}-61781420250 x^{5}-45081841167 x^{3}-31117061250 x^{6}-6715912500 x^{7}-\frac {89469521196}{5} x^{2}-\frac {59178013234}{15} x -\frac {1863616801}{5}}{\left (2+3 x \right )^{7} \left (3+5 x \right )}+9212500 \ln \left (2+3 x \right )-9212500 \ln \left (3+5 x \right )\) | \(69\) |
default | \(-\frac {7}{3 \left (2+3 x \right )^{7}}-\frac {77}{3 \left (2+3 x \right )^{6}}-\frac {1133}{5 \left (2+3 x \right )^{5}}-\frac {1870}{\left (2+3 x \right )^{4}}-\frac {46475}{3 \left (2+3 x \right )^{3}}-\frac {138875}{\left (2+3 x \right )^{2}}-\frac {1615625}{2+3 x}-\frac {378125}{3+5 x}+9212500 \ln \left (2+3 x \right )-9212500 \ln \left (3+5 x \right )\) | \(90\) |
parallelrisch | \(\frac {1132032000320 x -439794432000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+1337212800000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-82638336000000 \ln \left (x +\frac {3}{5}\right ) x +439794432000000 \ln \left (\frac {2}{3}+x \right ) x^{2}+82638336000000 \ln \left (\frac {2}{3}+x \right ) x +206834256779436 x^{5}+187516192323078 x^{6}+94433003186391 x^{7}+54332295114960 x^{3}+136866250079640 x^{4}+11980672000800 x^{2}+20378649718935 x^{8}+2540704320000000 \ln \left (\frac {2}{3}+x \right ) x^{4}+6792192000000 \ln \left (\frac {2}{3}+x \right )+1018669608000000 \ln \left (\frac {2}{3}+x \right ) x^{7}-1018669608000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-6792192000000 \ln \left (x +\frac {3}{5}\right )+3088961568000000 \ln \left (\frac {2}{3}+x \right ) x^{5}-1337212800000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-3088961568000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-2540704320000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+2346808464000000 \ln \left (\frac {2}{3}+x \right ) x^{6}-2346808464000000 \ln \left (x +\frac {3}{5}\right ) x^{6}+193418280000000 \ln \left (\frac {2}{3}+x \right ) x^{8}-193418280000000 \ln \left (x +\frac {3}{5}\right ) x^{8}}{1920 \left (2+3 x \right )^{7} \left (3+5 x \right )}\) | \(208\) |
(-68137326675*x^4-61781420250*x^5-45081841167*x^3-31117061250*x^6-67159125 00*x^7-89469521196/5*x^2-59178013234/15*x-1863616801/5)/(2+3*x)^7/(3+5*x)+ 9212500*ln(2+3*x)-9212500*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.80 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {100738687500 \, x^{7} + 466755918750 \, x^{6} + 926721303750 \, x^{5} + 1022059900125 \, x^{4} + 676227617505 \, x^{3} + 268408563588 \, x^{2} + 138187500 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )} \log \left (5 \, x + 3\right ) - 138187500 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )} \log \left (3 \, x + 2\right ) + 59178013234 \, x + 5590850403}{15 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )}} \]
-1/15*(100738687500*x^7 + 466755918750*x^6 + 926721303750*x^5 + 1022059900 125*x^4 + 676227617505*x^3 + 268408563588*x^2 + 138187500*(10935*x^8 + 575 91*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x^3 + 24864*x^2 + 46 72*x + 384)*log(5*x + 3) - 138187500*(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384)*log(3*x + 2) + 59178013234*x + 5590850403)/(10935*x^8 + 57591*x^7 + 132678*x^6 + 17 4636*x^5 + 143640*x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384)
Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=\frac {- 100738687500 x^{7} - 466755918750 x^{6} - 926721303750 x^{5} - 1022059900125 x^{4} - 676227617505 x^{3} - 268408563588 x^{2} - 59178013234 x - 5590850403}{164025 x^{8} + 863865 x^{7} + 1990170 x^{6} + 2619540 x^{5} + 2154600 x^{4} + 1134000 x^{3} + 372960 x^{2} + 70080 x + 5760} - 9212500 \log {\left (x + \frac {3}{5} \right )} + 9212500 \log {\left (x + \frac {2}{3} \right )} \]
(-100738687500*x**7 - 466755918750*x**6 - 926721303750*x**5 - 102205990012 5*x**4 - 676227617505*x**3 - 268408563588*x**2 - 59178013234*x - 559085040 3)/(164025*x**8 + 863865*x**7 + 1990170*x**6 + 2619540*x**5 + 2154600*x**4 + 1134000*x**3 + 372960*x**2 + 70080*x + 5760) - 9212500*log(x + 3/5) + 9 212500*log(x + 2/3)
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {100738687500 \, x^{7} + 466755918750 \, x^{6} + 926721303750 \, x^{5} + 1022059900125 \, x^{4} + 676227617505 \, x^{3} + 268408563588 \, x^{2} + 59178013234 \, x + 5590850403}{15 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )}} - 9212500 \, \log \left (5 \, x + 3\right ) + 9212500 \, \log \left (3 \, x + 2\right ) \]
-1/15*(100738687500*x^7 + 466755918750*x^6 + 926721303750*x^5 + 1022059900 125*x^4 + 676227617505*x^3 + 268408563588*x^2 + 59178013234*x + 5590850403 )/(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x^ 3 + 24864*x^2 + 4672*x + 384) - 9212500*log(5*x + 3) + 9212500*log(3*x + 2 )
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {378125}{5 \, x + 3} + \frac {625 \, {\left (\frac {120779019}{5 \, x + 3} + \frac {110006829}{{\left (5 \, x + 3\right )}^{2}} + \frac {54129465}{{\left (5 \, x + 3\right )}^{3}} + \frac {15246900}{{\left (5 \, x + 3\right )}^{4}} + \frac {2349450}{{\left (5 \, x + 3\right )}^{5}} + \frac {157100}{{\left (5 \, x + 3\right )}^{6}} + 55800576\right )}}{{\left (\frac {1}{5 \, x + 3} + 3\right )}^{7}} + 9212500 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]
-378125/(5*x + 3) + 625*(120779019/(5*x + 3) + 110006829/(5*x + 3)^2 + 541 29465/(5*x + 3)^3 + 15246900/(5*x + 3)^4 + 2349450/(5*x + 3)^5 + 157100/(5 *x + 3)^6 + 55800576)/(1/(5*x + 3) + 3)^7 + 9212500*log(abs(-1/(5*x + 3) - 3))
Time = 1.37 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=18425000\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {1842500\,x^7}{3}+\frac {25610750\,x^6}{9}+\frac {457640150\,x^5}{81}+\frac {1514162815\,x^4}{243}+\frac {1669697821\,x^3}{405}+\frac {29823173732\,x^2}{18225}+\frac {59178013234\,x}{164025}+\frac {1863616801}{54675}}{x^8+\frac {79\,x^7}{15}+\frac {182\,x^6}{15}+\frac {2156\,x^5}{135}+\frac {1064\,x^4}{81}+\frac {560\,x^3}{81}+\frac {8288\,x^2}{3645}+\frac {4672\,x}{10935}+\frac {128}{3645}} \]
18425000*atanh(30*x + 19) - ((59178013234*x)/164025 + (29823173732*x^2)/18 225 + (1669697821*x^3)/405 + (1514162815*x^4)/243 + (457640150*x^5)/81 + ( 25610750*x^6)/9 + (1842500*x^7)/3 + 1863616801/54675)/((4672*x)/10935 + (8 288*x^2)/3645 + (560*x^3)/81 + (1064*x^4)/81 + (2156*x^5)/135 + (182*x^6)/ 15 + (79*x^7)/15 + x^8 + 128/3645)