Integrand size = 84, antiderivative size = 26 \[ \int \frac {288 x^2+56 x^3+\left (-864 x^2-112 x^3\right ) \log (x)+\left (-2592-1008 x-98 x^2\right ) \log ^2(x)}{16 x^6+\left (288 x^4+56 x^5\right ) \log (x)+\left (1296 x^2+504 x^3+49 x^4\right ) \log ^2(x)} \, dx=\frac {2}{x \left (1+\frac {x^2}{\left (9+\frac {7 x}{4}\right ) \log (x)}\right )} \] Output:
2/(1+x^2/ln(x)/(9+7/4*x))/x
Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {288 x^2+56 x^3+\left (-864 x^2-112 x^3\right ) \log (x)+\left (-2592-1008 x-98 x^2\right ) \log ^2(x)}{16 x^6+\left (288 x^4+56 x^5\right ) \log (x)+\left (1296 x^2+504 x^3+49 x^4\right ) \log ^2(x)} \, dx=-2 \left (-\frac {1}{x}+\frac {4 x}{4 x^2+36 \log (x)+7 x \log (x)}\right ) \] Input:
Integrate[(288*x^2 + 56*x^3 + (-864*x^2 - 112*x^3)*Log[x] + (-2592 - 1008* x - 98*x^2)*Log[x]^2)/(16*x^6 + (288*x^4 + 56*x^5)*Log[x] + (1296*x^2 + 50 4*x^3 + 49*x^4)*Log[x]^2),x]
Output:
-2*(-x^(-1) + (4*x)/(4*x^2 + 36*Log[x] + 7*x*Log[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.
\(\displaystyle \int \frac {56 x^3+288 x^2+\left (-98 x^2-1008 x-2592\right ) \log ^2(x)+\left (-112 x^3-864 x^2\right ) \log (x)}{16 x^6+\left (56 x^5+288 x^4\right ) \log (x)+\left (49 x^4+504 x^3+1296 x^2\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {56 x^3+288 x^2+\left (-98 x^2-1008 x-2592\right ) \log ^2(x)+\left (-112 x^3-864 x^2\right ) \log (x)}{x^2 \left (4 x^2+7 x \log (x)+36 \log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2}{x^2}-\frac {288}{(7 x+36) \left (4 x^2+7 x \log (x)+36 \log (x)\right )}+\frac {8 \left (28 x^3+337 x^2+504 x+1296\right )}{(7 x+36) \left (4 x^2+7 x \log (x)+36 \log (x)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {27360}{49} \int \frac {1}{\left (4 x^2+7 \log (x) x+36 \log (x)\right )^2}dx+\frac {1544}{7} \int \frac {x}{\left (4 x^2+7 \log (x) x+36 \log (x)\right )^2}dx+32 \int \frac {x^2}{\left (4 x^2+7 \log (x) x+36 \log (x)\right )^2}dx+\frac {1492992}{49} \int \frac {1}{(7 x+36) \left (4 x^2+7 \log (x) x+36 \log (x)\right )^2}dx-288 \int \frac {1}{(7 x+36) \left (4 x^2+7 \log (x) x+36 \log (x)\right )}dx+\frac {2}{x}\) |
Input:
Int[(288*x^2 + 56*x^3 + (-864*x^2 - 112*x^3)*Log[x] + (-2592 - 1008*x - 98 *x^2)*Log[x]^2)/(16*x^6 + (288*x^4 + 56*x^5)*Log[x] + (1296*x^2 + 504*x^3 + 49*x^4)*Log[x]^2),x]
Output:
$Aborted
Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {2}{x}-\frac {8 x}{7 x \ln \left (x \right )+4 x^{2}+36 \ln \left (x \right )}\) | \(27\) |
norman | \(\frac {14 x \ln \left (x \right )+72 \ln \left (x \right )}{x \left (7 x \ln \left (x \right )+4 x^{2}+36 \ln \left (x \right )\right )}\) | \(32\) |
default | \(-\frac {2 \left (-7 x \ln \left (x \right )-36 \ln \left (x \right )\right )}{x \left (7 x \ln \left (x \right )+4 x^{2}+36 \ln \left (x \right )\right )}\) | \(33\) |
parallelrisch | \(\frac {98 x \ln \left (x \right )+504 \ln \left (x \right )}{7 x \left (7 x \ln \left (x \right )+4 x^{2}+36 \ln \left (x \right )\right )}\) | \(33\) |
Input:
int(((-98*x^2-1008*x-2592)*ln(x)^2+(-112*x^3-864*x^2)*ln(x)+56*x^3+288*x^2 )/((49*x^4+504*x^3+1296*x^2)*ln(x)^2+(56*x^5+288*x^4)*ln(x)+16*x^6),x,meth od=_RETURNVERBOSE)
Output:
2/x-8*x/(7*x*ln(x)+4*x^2+36*ln(x))
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {288 x^2+56 x^3+\left (-864 x^2-112 x^3\right ) \log (x)+\left (-2592-1008 x-98 x^2\right ) \log ^2(x)}{16 x^6+\left (288 x^4+56 x^5\right ) \log (x)+\left (1296 x^2+504 x^3+49 x^4\right ) \log ^2(x)} \, dx=\frac {2 \, {\left (7 \, x + 36\right )} \log \left (x\right )}{4 \, x^{3} + {\left (7 \, x^{2} + 36 \, x\right )} \log \left (x\right )} \] Input:
integrate(((-98*x^2-1008*x-2592)*log(x)^2+(-112*x^3-864*x^2)*log(x)+56*x^3 +288*x^2)/((49*x^4+504*x^3+1296*x^2)*log(x)^2+(56*x^5+288*x^4)*log(x)+16*x ^6),x, algorithm="fricas")
Output:
2*(7*x + 36)*log(x)/(4*x^3 + (7*x^2 + 36*x)*log(x))
Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {288 x^2+56 x^3+\left (-864 x^2-112 x^3\right ) \log (x)+\left (-2592-1008 x-98 x^2\right ) \log ^2(x)}{16 x^6+\left (288 x^4+56 x^5\right ) \log (x)+\left (1296 x^2+504 x^3+49 x^4\right ) \log ^2(x)} \, dx=- \frac {8 x}{4 x^{2} + \left (7 x + 36\right ) \log {\left (x \right )}} + \frac {2}{x} \] Input:
integrate(((-98*x**2-1008*x-2592)*ln(x)**2+(-112*x**3-864*x**2)*ln(x)+56*x **3+288*x**2)/((49*x**4+504*x**3+1296*x**2)*ln(x)**2+(56*x**5+288*x**4)*ln (x)+16*x**6),x)
Output:
-8*x/(4*x**2 + (7*x + 36)*log(x)) + 2/x
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {288 x^2+56 x^3+\left (-864 x^2-112 x^3\right ) \log (x)+\left (-2592-1008 x-98 x^2\right ) \log ^2(x)}{16 x^6+\left (288 x^4+56 x^5\right ) \log (x)+\left (1296 x^2+504 x^3+49 x^4\right ) \log ^2(x)} \, dx=\frac {2 \, {\left (7 \, x + 36\right )} \log \left (x\right )}{4 \, x^{3} + {\left (7 \, x^{2} + 36 \, x\right )} \log \left (x\right )} \] Input:
integrate(((-98*x^2-1008*x-2592)*log(x)^2+(-112*x^3-864*x^2)*log(x)+56*x^3 +288*x^2)/((49*x^4+504*x^3+1296*x^2)*log(x)^2+(56*x^5+288*x^4)*log(x)+16*x ^6),x, algorithm="maxima")
Output:
2*(7*x + 36)*log(x)/(4*x^3 + (7*x^2 + 36*x)*log(x))
Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {288 x^2+56 x^3+\left (-864 x^2-112 x^3\right ) \log (x)+\left (-2592-1008 x-98 x^2\right ) \log ^2(x)}{16 x^6+\left (288 x^4+56 x^5\right ) \log (x)+\left (1296 x^2+504 x^3+49 x^4\right ) \log ^2(x)} \, dx=-\frac {8 \, x}{4 \, x^{2} + 7 \, x \log \left (x\right ) + 36 \, \log \left (x\right )} + \frac {2}{x} \] Input:
integrate(((-98*x^2-1008*x-2592)*log(x)^2+(-112*x^3-864*x^2)*log(x)+56*x^3 +288*x^2)/((49*x^4+504*x^3+1296*x^2)*log(x)^2+(56*x^5+288*x^4)*log(x)+16*x ^6),x, algorithm="giac")
Output:
-8*x/(4*x^2 + 7*x*log(x) + 36*log(x)) + 2/x
Time = 1.93 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {288 x^2+56 x^3+\left (-864 x^2-112 x^3\right ) \log (x)+\left (-2592-1008 x-98 x^2\right ) \log ^2(x)}{16 x^6+\left (288 x^4+56 x^5\right ) \log (x)+\left (1296 x^2+504 x^3+49 x^4\right ) \log ^2(x)} \, dx=\frac {2\,\ln \left (x\right )\,\left (7\,x+36\right )}{x\,\left (36\,\ln \left (x\right )+7\,x\,\ln \left (x\right )+4\,x^2\right )} \] Input:
int(-(log(x)*(864*x^2 + 112*x^3) + log(x)^2*(1008*x + 98*x^2 + 2592) - 288 *x^2 - 56*x^3)/(log(x)*(288*x^4 + 56*x^5) + log(x)^2*(1296*x^2 + 504*x^3 + 49*x^4) + 16*x^6),x)
Output:
(2*log(x)*(7*x + 36))/(x*(36*log(x) + 7*x*log(x) + 4*x^2))
Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {288 x^2+56 x^3+\left (-864 x^2-112 x^3\right ) \log (x)+\left (-2592-1008 x-98 x^2\right ) \log ^2(x)}{16 x^6+\left (288 x^4+56 x^5\right ) \log (x)+\left (1296 x^2+504 x^3+49 x^4\right ) \log ^2(x)} \, dx=\frac {2 \,\mathrm {log}\left (x \right ) \left (7 x +36\right )}{x \left (7 \,\mathrm {log}\left (x \right ) x +36 \,\mathrm {log}\left (x \right )+4 x^{2}\right )} \] Input:
int(((-98*x^2-1008*x-2592)*log(x)^2+(-112*x^3-864*x^2)*log(x)+56*x^3+288*x ^2)/((49*x^4+504*x^3+1296*x^2)*log(x)^2+(56*x^5+288*x^4)*log(x)+16*x^6),x)
Output:
(2*log(x)*(7*x + 36))/(x*(7*log(x)*x + 36*log(x) + 4*x**2))