Integrand size = 221, antiderivative size = 25 \[ \int \frac {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5+\left (3456+864 x-1010 x^2+194 x^3-12 x^4\right ) \log (4+3 x)+\left (864+360 x-192 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{5184+720 x-1604 x^2+491 x^3-62 x^4+3 x^5+\left (3456+960 x-952 x^2+188 x^3-12 x^4\right ) \log (4+3 x)+\left (864+368 x-186 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)} \, dx=\frac {x^2}{x+\frac {x^2}{(-6+x-\log (4+3 x))^2}} \]
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5+\left (3456+864 x-1010 x^2+194 x^3-12 x^4\right ) \log (4+3 x)+\left (864+360 x-192 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{5184+720 x-1604 x^2+491 x^3-62 x^4+3 x^5+\left (3456+960 x-952 x^2+188 x^3-12 x^4\right ) \log (4+3 x)+\left (864+368 x-186 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)} \, dx=\frac {x (6-x+\log (4+3 x))^2}{36-11 x+x^2-2 (-6+x) \log (4+3 x)+\log ^2(4+3 x)} \]
Integrate[(5184 + 432*x - 1740*x^2 + 518*x^3 - 62*x^4 + 3*x^5 + (3456 + 86 4*x - 1010*x^2 + 194*x^3 - 12*x^4)*Log[4 + 3*x] + (864 + 360*x - 192*x^2 + 18*x^3)*Log[4 + 3*x]^2 + (96 + 56*x - 12*x^2)*Log[4 + 3*x]^3 + (4 + 3*x)* Log[4 + 3*x]^4)/(5184 + 720*x - 1604*x^2 + 491*x^3 - 62*x^4 + 3*x^5 + (345 6 + 960*x - 952*x^2 + 188*x^3 - 12*x^4)*Log[4 + 3*x] + (864 + 368*x - 186* x^2 + 18*x^3)*Log[4 + 3*x]^2 + (96 + 56*x - 12*x^2)*Log[4 + 3*x]^3 + (4 + 3*x)*Log[4 + 3*x]^4),x]
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.
| \(\displaystyle \int \frac {3 x^5-62 x^4+518 x^3-1740 x^2+\left (-12 x^2+56 x+96\right ) \log ^3(3 x+4)+\left (18 x^3-192 x^2+360 x+864\right ) \log ^2(3 x+4)+\left (-12 x^4+194 x^3-1010 x^2+864 x+3456\right ) \log (3 x+4)+432 x+(3 x+4) \log ^4(3 x+4)+5184}{3 x^5-62 x^4+491 x^3-1604 x^2+\left (-12 x^2+56 x+96\right ) \log ^3(3 x+4)+\left (18 x^3-186 x^2+368 x+864\right ) \log ^2(3 x+4)+\left (-12 x^4+188 x^3-952 x^2+960 x+3456\right ) \log (3 x+4)+720 x+(3 x+4) \log ^4(3 x+4)+5184} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {3 x^5-62 x^4+518 x^3-1740 x^2+\left (-12 x^2+56 x+96\right ) \log ^3(3 x+4)+\left (18 x^3-192 x^2+360 x+864\right ) \log ^2(3 x+4)+\left (-12 x^4+194 x^3-1010 x^2+864 x+3456\right ) \log (3 x+4)+432 x+(3 x+4) \log ^4(3 x+4)+5184}{(3 x+4) \left (x^2-11 x+\log ^2(3 x+4)-2 x \log (3 x+4)+12 \log (3 x+4)+36\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2 \left (6 x^2-31 x-6 x \log (3 x+4)-2 \log (3 x+4)-8\right )}{(3 x+4) \left (x^2-11 x+\log ^2(3 x+4)-2 x \log (3 x+4)+12 \log (3 x+4)+36\right )^2}-\frac {2 x}{x^2-11 x+\log ^2(3 x+4)-2 x \log (3 x+4)+12 \log (3 x+4)+36}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {32}{9} \int \frac {1}{\left (x^2-2 \log (3 x+4) x-11 x+\log ^2(3 x+4)+12 \log (3 x+4)+36\right )^2}dx+\frac {100}{9} \int \frac {x}{\left (x^2-2 \log (3 x+4) x-11 x+\log ^2(3 x+4)+12 \log (3 x+4)+36\right )^2}dx-13 \int \frac {x^2}{\left (x^2-2 \log (3 x+4) x-11 x+\log ^2(3 x+4)+12 \log (3 x+4)+36\right )^2}dx+\frac {8}{9} \int \frac {\log (3 x+4)}{\left (x^2-2 \log (3 x+4) x-11 x+\log ^2(3 x+4)+12 \log (3 x+4)+36\right )^2}dx+2 \int \frac {x \log (3 x+4)}{\left (x^2-2 \log (3 x+4) x-11 x+\log ^2(3 x+4)+12 \log (3 x+4)+36\right )^2}dx-2 \int \frac {x^2 \log (3 x+4)}{\left (x^2-2 \log (3 x+4) x-11 x+\log ^2(3 x+4)+12 \log (3 x+4)+36\right )^2}dx-2 \int \frac {x}{x^2-2 \log (3 x+4) x-11 x+\log ^2(3 x+4)+12 \log (3 x+4)+36}dx+2 \int \frac {x^3}{\left (x^2-2 \log (3 x+4) x-11 x+\log ^2(3 x+4)+12 \log (3 x+4)+36\right )^2}dx-\frac {16}{9 \left (x^2-11 x+\log ^2(3 x+4)-2 x \log (3 x+4)+12 \log (3 x+4)+36\right )}+x\) |
Int[(5184 + 432*x - 1740*x^2 + 518*x^3 - 62*x^4 + 3*x^5 + (3456 + 864*x - 1010*x^2 + 194*x^3 - 12*x^4)*Log[4 + 3*x] + (864 + 360*x - 192*x^2 + 18*x^ 3)*Log[4 + 3*x]^2 + (96 + 56*x - 12*x^2)*Log[4 + 3*x]^3 + (4 + 3*x)*Log[4 + 3*x]^4)/(5184 + 720*x - 1604*x^2 + 491*x^3 - 62*x^4 + 3*x^5 + (3456 + 96 0*x - 952*x^2 + 188*x^3 - 12*x^4)*Log[4 + 3*x] + (864 + 368*x - 186*x^2 + 18*x^3)*Log[4 + 3*x]^2 + (96 + 56*x - 12*x^2)*Log[4 + 3*x]^3 + (4 + 3*x)*L og[4 + 3*x]^4),x]
3.17.93.3.1 Defintions of rubi rules used
Time = 0.33 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72
| method | result | size |
| risch | \(x -\frac {x^{2}}{\ln \left (4+3 x \right )^{2}-2 \ln \left (4+3 x \right ) x +x^{2}+12 \ln \left (4+3 x \right )-11 x +36}\) | \(43\) |
| parallelrisch | \(\frac {-864+588 x -288 \ln \left (4+3 x \right )+9 x^{3}-132 x^{2}+9 \ln \left (4+3 x \right )^{2} x -18 \ln \left (4+3 x \right ) x^{2}+156 \ln \left (4+3 x \right ) x -24 \ln \left (4+3 x \right )^{2}}{9 \ln \left (4+3 x \right )^{2}-18 \ln \left (4+3 x \right ) x +9 x^{2}+108 \ln \left (4+3 x \right )-99 x +324}\) | \(102\) |
| derivativedivides | \(\frac {\left (4+3 x \right )^{3}-44 \left (4+3 x \right )^{2}+1936+1488 x +132 \ln \left (4+3 x \right ) \left (4+3 x \right )-6 \ln \left (4+3 x \right ) \left (4+3 x \right )^{2}+9 \ln \left (4+3 x \right )^{2} \left (4+3 x \right )}{3 \left (4+3 x \right )^{2}-18 \ln \left (4+3 x \right ) \left (4+3 x \right )+27 \ln \left (4+3 x \right )^{2}+924-369 x +396 \ln \left (4+3 x \right )}\) | \(112\) |
| default | \(\frac {\left (4+3 x \right )^{3}-44 \left (4+3 x \right )^{2}+1936+1488 x +132 \ln \left (4+3 x \right ) \left (4+3 x \right )-6 \ln \left (4+3 x \right ) \left (4+3 x \right )^{2}+9 \ln \left (4+3 x \right )^{2} \left (4+3 x \right )}{3 \left (4+3 x \right )^{2}-18 \ln \left (4+3 x \right ) \left (4+3 x \right )+27 \ln \left (4+3 x \right )^{2}+924-369 x +396 \ln \left (4+3 x \right )}\) | \(112\) |
int(((4+3*x)*ln(4+3*x)^4+(-12*x^2+56*x+96)*ln(4+3*x)^3+(18*x^3-192*x^2+360 *x+864)*ln(4+3*x)^2+(-12*x^4+194*x^3-1010*x^2+864*x+3456)*ln(4+3*x)+3*x^5- 62*x^4+518*x^3-1740*x^2+432*x+5184)/((4+3*x)*ln(4+3*x)^4+(-12*x^2+56*x+96) *ln(4+3*x)^3+(18*x^3-186*x^2+368*x+864)*ln(4+3*x)^2+(-12*x^4+188*x^3-952*x ^2+960*x+3456)*ln(4+3*x)+3*x^5-62*x^4+491*x^3-1604*x^2+720*x+5184),x,metho d=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (25) = 50\).
Time = 0.36 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5+\left (3456+864 x-1010 x^2+194 x^3-12 x^4\right ) \log (4+3 x)+\left (864+360 x-192 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{5184+720 x-1604 x^2+491 x^3-62 x^4+3 x^5+\left (3456+960 x-952 x^2+188 x^3-12 x^4\right ) \log (4+3 x)+\left (864+368 x-186 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)} \, dx=\frac {x^{3} + x \log \left (3 \, x + 4\right )^{2} - 12 \, x^{2} - 2 \, {\left (x^{2} - 6 \, x\right )} \log \left (3 \, x + 4\right ) + 36 \, x}{x^{2} - 2 \, {\left (x - 6\right )} \log \left (3 \, x + 4\right ) + \log \left (3 \, x + 4\right )^{2} - 11 \, x + 36} \]
integrate(((4+3*x)*log(4+3*x)^4+(-12*x^2+56*x+96)*log(4+3*x)^3+(18*x^3-192 *x^2+360*x+864)*log(4+3*x)^2+(-12*x^4+194*x^3-1010*x^2+864*x+3456)*log(4+3 *x)+3*x^5-62*x^4+518*x^3-1740*x^2+432*x+5184)/((4+3*x)*log(4+3*x)^4+(-12*x ^2+56*x+96)*log(4+3*x)^3+(18*x^3-186*x^2+368*x+864)*log(4+3*x)^2+(-12*x^4+ 188*x^3-952*x^2+960*x+3456)*log(4+3*x)+3*x^5-62*x^4+491*x^3-1604*x^2+720*x +5184),x, algorithm=\
(x^3 + x*log(3*x + 4)^2 - 12*x^2 - 2*(x^2 - 6*x)*log(3*x + 4) + 36*x)/(x^2 - 2*(x - 6)*log(3*x + 4) + log(3*x + 4)^2 - 11*x + 36)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5+\left (3456+864 x-1010 x^2+194 x^3-12 x^4\right ) \log (4+3 x)+\left (864+360 x-192 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{5184+720 x-1604 x^2+491 x^3-62 x^4+3 x^5+\left (3456+960 x-952 x^2+188 x^3-12 x^4\right ) \log (4+3 x)+\left (864+368 x-186 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)} \, dx=- \frac {x^{2}}{x^{2} - 11 x + \left (12 - 2 x\right ) \log {\left (3 x + 4 \right )} + \log {\left (3 x + 4 \right )}^{2} + 36} + x \]
integrate(((4+3*x)*ln(4+3*x)**4+(-12*x**2+56*x+96)*ln(4+3*x)**3+(18*x**3-1 92*x**2+360*x+864)*ln(4+3*x)**2+(-12*x**4+194*x**3-1010*x**2+864*x+3456)*l n(4+3*x)+3*x**5-62*x**4+518*x**3-1740*x**2+432*x+5184)/((4+3*x)*ln(4+3*x)* *4+(-12*x**2+56*x+96)*ln(4+3*x)**3+(18*x**3-186*x**2+368*x+864)*ln(4+3*x)* *2+(-12*x**4+188*x**3-952*x**2+960*x+3456)*ln(4+3*x)+3*x**5-62*x**4+491*x* *3-1604*x**2+720*x+5184),x)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5+\left (3456+864 x-1010 x^2+194 x^3-12 x^4\right ) \log (4+3 x)+\left (864+360 x-192 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{5184+720 x-1604 x^2+491 x^3-62 x^4+3 x^5+\left (3456+960 x-952 x^2+188 x^3-12 x^4\right ) \log (4+3 x)+\left (864+368 x-186 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)} \, dx=\frac {x^{3} + x \log \left (3 \, x + 4\right )^{2} - 12 \, x^{2} - 2 \, {\left (x^{2} - 6 \, x\right )} \log \left (3 \, x + 4\right ) + 36 \, x}{x^{2} - 2 \, {\left (x - 6\right )} \log \left (3 \, x + 4\right ) + \log \left (3 \, x + 4\right )^{2} - 11 \, x + 36} \]
integrate(((4+3*x)*log(4+3*x)^4+(-12*x^2+56*x+96)*log(4+3*x)^3+(18*x^3-192 *x^2+360*x+864)*log(4+3*x)^2+(-12*x^4+194*x^3-1010*x^2+864*x+3456)*log(4+3 *x)+3*x^5-62*x^4+518*x^3-1740*x^2+432*x+5184)/((4+3*x)*log(4+3*x)^4+(-12*x ^2+56*x+96)*log(4+3*x)^3+(18*x^3-186*x^2+368*x+864)*log(4+3*x)^2+(-12*x^4+ 188*x^3-952*x^2+960*x+3456)*log(4+3*x)+3*x^5-62*x^4+491*x^3-1604*x^2+720*x +5184),x, algorithm=\
(x^3 + x*log(3*x + 4)^2 - 12*x^2 - 2*(x^2 - 6*x)*log(3*x + 4) + 36*x)/(x^2 - 2*(x - 6)*log(3*x + 4) + log(3*x + 4)^2 - 11*x + 36)
Time = 0.35 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5+\left (3456+864 x-1010 x^2+194 x^3-12 x^4\right ) \log (4+3 x)+\left (864+360 x-192 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{5184+720 x-1604 x^2+491 x^3-62 x^4+3 x^5+\left (3456+960 x-952 x^2+188 x^3-12 x^4\right ) \log (4+3 x)+\left (864+368 x-186 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)} \, dx=x - \frac {x^{2}}{x^{2} - 2 \, x \log \left (3 \, x + 4\right ) + \log \left (3 \, x + 4\right )^{2} - 11 \, x + 12 \, \log \left (3 \, x + 4\right ) + 36} \]
integrate(((4+3*x)*log(4+3*x)^4+(-12*x^2+56*x+96)*log(4+3*x)^3+(18*x^3-192 *x^2+360*x+864)*log(4+3*x)^2+(-12*x^4+194*x^3-1010*x^2+864*x+3456)*log(4+3 *x)+3*x^5-62*x^4+518*x^3-1740*x^2+432*x+5184)/((4+3*x)*log(4+3*x)^4+(-12*x ^2+56*x+96)*log(4+3*x)^3+(18*x^3-186*x^2+368*x+864)*log(4+3*x)^2+(-12*x^4+ 188*x^3-952*x^2+960*x+3456)*log(4+3*x)+3*x^5-62*x^4+491*x^3-1604*x^2+720*x +5184),x, algorithm=\
Timed out. \[ \int \frac {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5+\left (3456+864 x-1010 x^2+194 x^3-12 x^4\right ) \log (4+3 x)+\left (864+360 x-192 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{5184+720 x-1604 x^2+491 x^3-62 x^4+3 x^5+\left (3456+960 x-952 x^2+188 x^3-12 x^4\right ) \log (4+3 x)+\left (864+368 x-186 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)} \, dx=\int \frac {432\,x+{\ln \left (3\,x+4\right )}^2\,\left (18\,x^3-192\,x^2+360\,x+864\right )+{\ln \left (3\,x+4\right )}^4\,\left (3\,x+4\right )+{\ln \left (3\,x+4\right )}^3\,\left (-12\,x^2+56\,x+96\right )-1740\,x^2+518\,x^3-62\,x^4+3\,x^5+\ln \left (3\,x+4\right )\,\left (-12\,x^4+194\,x^3-1010\,x^2+864\,x+3456\right )+5184}{720\,x+{\ln \left (3\,x+4\right )}^2\,\left (18\,x^3-186\,x^2+368\,x+864\right )+{\ln \left (3\,x+4\right )}^4\,\left (3\,x+4\right )+{\ln \left (3\,x+4\right )}^3\,\left (-12\,x^2+56\,x+96\right )-1604\,x^2+491\,x^3-62\,x^4+3\,x^5+\ln \left (3\,x+4\right )\,\left (-12\,x^4+188\,x^3-952\,x^2+960\,x+3456\right )+5184} \,d x \]
int((432*x + log(3*x + 4)^2*(360*x - 192*x^2 + 18*x^3 + 864) + log(3*x + 4 )^4*(3*x + 4) + log(3*x + 4)^3*(56*x - 12*x^2 + 96) - 1740*x^2 + 518*x^3 - 62*x^4 + 3*x^5 + log(3*x + 4)*(864*x - 1010*x^2 + 194*x^3 - 12*x^4 + 3456 ) + 5184)/(720*x + log(3*x + 4)^2*(368*x - 186*x^2 + 18*x^3 + 864) + log(3 *x + 4)^4*(3*x + 4) + log(3*x + 4)^3*(56*x - 12*x^2 + 96) - 1604*x^2 + 491 *x^3 - 62*x^4 + 3*x^5 + log(3*x + 4)*(960*x - 952*x^2 + 188*x^3 - 12*x^4 + 3456) + 5184),x)
int((432*x + log(3*x + 4)^2*(360*x - 192*x^2 + 18*x^3 + 864) + log(3*x + 4 )^4*(3*x + 4) + log(3*x + 4)^3*(56*x - 12*x^2 + 96) - 1740*x^2 + 518*x^3 - 62*x^4 + 3*x^5 + log(3*x + 4)*(864*x - 1010*x^2 + 194*x^3 - 12*x^4 + 3456 ) + 5184)/(720*x + log(3*x + 4)^2*(368*x - 186*x^2 + 18*x^3 + 864) + log(3 *x + 4)^4*(3*x + 4) + log(3*x + 4)^3*(56*x - 12*x^2 + 96) - 1604*x^2 + 491 *x^3 - 62*x^4 + 3*x^5 + log(3*x + 4)*(960*x - 952*x^2 + 188*x^3 - 12*x^4 + 3456) + 5184), x)