Integrand size = 78, antiderivative size = 32 \[ \int \frac {54-104 x-24 x^2+e^x \left (-2 x-56 x^2-8 x^3\right )+\left (56+16 x+e^x \left (56 x+8 x^2\right )\right ) \log (x)}{-x-28 x^2-4 x^3+\left (28 x+4 x^2\right ) \log (x)} \, dx=\log \left (\frac {1}{25} e^{2 e^x} x^2 \left (\frac {1}{4}-(7+x) (-x+\log (x))\right )^2\right ) \]
Time = 0.51 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {54-104 x-24 x^2+e^x \left (-2 x-56 x^2-8 x^3\right )+\left (56+16 x+e^x \left (56 x+8 x^2\right )\right ) \log (x)}{-x-28 x^2-4 x^3+\left (28 x+4 x^2\right ) \log (x)} \, dx=2 \left (e^x+\log (x)+\log \left (1+28 x+4 x^2-4 (7+x) \log (x)\right )\right ) \]
Integrate[(54 - 104*x - 24*x^2 + E^x*(-2*x - 56*x^2 - 8*x^3) + (56 + 16*x + E^x*(56*x + 8*x^2))*Log[x])/(-x - 28*x^2 - 4*x^3 + (28*x + 4*x^2)*Log[x] ),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.
\(\displaystyle \int \frac {-24 x^2+\left (e^x \left (8 x^2+56 x\right )+16 x+56\right ) \log (x)+e^x \left (-8 x^3-56 x^2-2 x\right )-104 x+54}{-4 x^3-28 x^2+\left (4 x^2+28 x\right ) \log (x)-x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {24 x}{4 x^2+28 x-4 x \log (x)-28 \log (x)+1}+\frac {104}{4 x^2+28 x-4 x \log (x)-28 \log (x)+1}+\frac {16 \log (x)}{-4 x^2-28 x+4 x \log (x)+28 \log (x)-1}-\frac {56 \log (x)}{x \left (4 x^2+28 x-4 x \log (x)-28 \log (x)+1\right )}-\frac {54}{x \left (4 x^2+28 x-4 x \log (x)-28 \log (x)+1\right )}+2 e^x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 48 \int \frac {1}{4 x^2-4 \log (x) x+28 x-28 \log (x)+1}dx-56 \int \frac {1}{x \left (4 x^2-4 \log (x) x+28 x-28 \log (x)+1\right )}dx+8 \int \frac {x}{4 x^2-4 \log (x) x+28 x-28 \log (x)+1}dx-2 \int \frac {1}{(x+7) \left (4 x^2-4 \log (x) x+28 x-28 \log (x)+1\right )}dx+2 e^x+2 \log (x)+2 \log (x+7)\) |
Int[(54 - 104*x - 24*x^2 + E^x*(-2*x - 56*x^2 - 8*x^3) + (56 + 16*x + E^x* (56*x + 8*x^2))*Log[x])/(-x - 28*x^2 - 4*x^3 + (28*x + 4*x^2)*Log[x]),x]
3.30.39.3.1 Defintions of rubi rules used
Time = 0.79 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94
method | result | size |
parallelrisch | \(2 \ln \left (x^{2}-x \ln \left (x \right )+7 x -7 \ln \left (x \right )+\frac {1}{4}\right )+2 \ln \left (x \right )+2 \,{\mathrm e}^{x}\) | \(30\) |
default | \(2 \ln \left (x \right )+2 \ln \left (4 x \ln \left (x \right )-4 x^{2}+28 \ln \left (x \right )-28 x -1\right )+2 \,{\mathrm e}^{x}\) | \(32\) |
norman | \(2 \ln \left (x \right )+2 \,{\mathrm e}^{x}+2 \ln \left (4 x^{2}-4 x \ln \left (x \right )+28 x -28 \ln \left (x \right )+1\right )\) | \(32\) |
parts | \(2 \ln \left (x \right )+2 \ln \left (4 x \ln \left (x \right )-4 x^{2}+28 \ln \left (x \right )-28 x -1\right )+2 \,{\mathrm e}^{x}\) | \(32\) |
risch | \(2 \ln \left (x^{2}+7 x \right )+2 \,{\mathrm e}^{x}+2 \ln \left (\ln \left (x \right )-\frac {4 x^{2}+28 x +1}{4 \left (x +7\right )}\right )\) | \(39\) |
int((((8*x^2+56*x)*exp(x)+16*x+56)*ln(x)+(-8*x^3-56*x^2-2*x)*exp(x)-24*x^2 -104*x+54)/((4*x^2+28*x)*ln(x)-4*x^3-28*x^2-x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {54-104 x-24 x^2+e^x \left (-2 x-56 x^2-8 x^3\right )+\left (56+16 x+e^x \left (56 x+8 x^2\right )\right ) \log (x)}{-x-28 x^2-4 x^3+\left (28 x+4 x^2\right ) \log (x)} \, dx=2 \, e^{x} + 2 \, \log \left (x^{2} + 7 \, x\right ) + 2 \, \log \left (-\frac {4 \, x^{2} - 4 \, {\left (x + 7\right )} \log \left (x\right ) + 28 \, x + 1}{x + 7}\right ) \]
integrate((((8*x^2+56*x)*exp(x)+16*x+56)*log(x)+(-8*x^3-56*x^2-2*x)*exp(x) -24*x^2-104*x+54)/((4*x^2+28*x)*log(x)-4*x^3-28*x^2-x),x, algorithm=\
Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {54-104 x-24 x^2+e^x \left (-2 x-56 x^2-8 x^3\right )+\left (56+16 x+e^x \left (56 x+8 x^2\right )\right ) \log (x)}{-x-28 x^2-4 x^3+\left (28 x+4 x^2\right ) \log (x)} \, dx=2 e^{x} + 2 \log {\left (x^{2} + 7 x \right )} + 2 \log {\left (\log {\left (x \right )} + \frac {- 4 x^{2} - 28 x - 1}{4 x + 28} \right )} \]
integrate((((8*x**2+56*x)*exp(x)+16*x+56)*ln(x)+(-8*x**3-56*x**2-2*x)*exp( x)-24*x**2-104*x+54)/((4*x**2+28*x)*ln(x)-4*x**3-28*x**2-x),x)
Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {54-104 x-24 x^2+e^x \left (-2 x-56 x^2-8 x^3\right )+\left (56+16 x+e^x \left (56 x+8 x^2\right )\right ) \log (x)}{-x-28 x^2-4 x^3+\left (28 x+4 x^2\right ) \log (x)} \, dx=2 \, e^{x} + 2 \, \log \left (x + 7\right ) + 2 \, \log \left (x\right ) + 2 \, \log \left (-\frac {4 \, x^{2} - 4 \, {\left (x + 7\right )} \log \left (x\right ) + 28 \, x + 1}{4 \, {\left (x + 7\right )}}\right ) \]
integrate((((8*x^2+56*x)*exp(x)+16*x+56)*log(x)+(-8*x^3-56*x^2-2*x)*exp(x) -24*x^2-104*x+54)/((4*x^2+28*x)*log(x)-4*x^3-28*x^2-x),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {54-104 x-24 x^2+e^x \left (-2 x-56 x^2-8 x^3\right )+\left (56+16 x+e^x \left (56 x+8 x^2\right )\right ) \log (x)}{-x-28 x^2-4 x^3+\left (28 x+4 x^2\right ) \log (x)} \, dx=2 \, e^{x} + 2 \, \log \left (-4 \, x^{2} + 4 \, x \log \left (x\right ) - 28 \, x + 28 \, \log \left (x\right ) - 1\right ) + 2 \, \log \left (x\right ) \]
integrate((((8*x^2+56*x)*exp(x)+16*x+56)*log(x)+(-8*x^3-56*x^2-2*x)*exp(x) -24*x^2-104*x+54)/((4*x^2+28*x)*log(x)-4*x^3-28*x^2-x),x, algorithm=\
Timed out. \[ \int \frac {54-104 x-24 x^2+e^x \left (-2 x-56 x^2-8 x^3\right )+\left (56+16 x+e^x \left (56 x+8 x^2\right )\right ) \log (x)}{-x-28 x^2-4 x^3+\left (28 x+4 x^2\right ) \log (x)} \, dx=\int \frac {104\,x-\ln \left (x\right )\,\left (16\,x+{\mathrm {e}}^x\,\left (8\,x^2+56\,x\right )+56\right )+24\,x^2+{\mathrm {e}}^x\,\left (8\,x^3+56\,x^2+2\,x\right )-54}{x-\ln \left (x\right )\,\left (4\,x^2+28\,x\right )+28\,x^2+4\,x^3} \,d x \]
int((104*x - log(x)*(16*x + exp(x)*(56*x + 8*x^2) + 56) + 24*x^2 + exp(x)* (2*x + 56*x^2 + 8*x^3) - 54)/(x - log(x)*(28*x + 4*x^2) + 28*x^2 + 4*x^3), x)