Integrand size = 159, antiderivative size = 25 \[ \int \frac {-115+122 x+26 x^2+15 x^3+3 x^4+\left (-27 x-36 x^2-6 x^3\right ) \log (x)+\left (15 x+3 x^2\right ) \log ^2(x)+\left (-15+12 x+3 x^2\right ) \log (5+x)}{-115 x^2+112 x^3+42 x^4+3 x^5+\left (115 x-217 x^2-78 x^3-6 x^4\right ) \log (x)+\left (105 x+36 x^2+3 x^3\right ) \log ^2(x)+\left (-15 x^2-3 x^3+\left (15 x+3 x^2\right ) \log (x)\right ) \log (5+x)} \, dx=\log \left (7+x-\frac {\frac {23}{3}-2 x+\log (5+x)}{x-\log (x)}\right ) \]
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-115+122 x+26 x^2+15 x^3+3 x^4+\left (-27 x-36 x^2-6 x^3\right ) \log (x)+\left (15 x+3 x^2\right ) \log ^2(x)+\left (-15+12 x+3 x^2\right ) \log (5+x)}{-115 x^2+112 x^3+42 x^4+3 x^5+\left (115 x-217 x^2-78 x^3-6 x^4\right ) \log (x)+\left (105 x+36 x^2+3 x^3\right ) \log ^2(x)+\left (-15 x^2-3 x^3+\left (15 x+3 x^2\right ) \log (x)\right ) \log (5+x)} \, dx=-\log (x-\log (x))+\log \left (23-27 x-3 x^2+21 \log (x)+3 x \log (x)+3 \log (5+x)\right ) \]
Integrate[(-115 + 122*x + 26*x^2 + 15*x^3 + 3*x^4 + (-27*x - 36*x^2 - 6*x^ 3)*Log[x] + (15*x + 3*x^2)*Log[x]^2 + (-15 + 12*x + 3*x^2)*Log[5 + x])/(-1 15*x^2 + 112*x^3 + 42*x^4 + 3*x^5 + (115*x - 217*x^2 - 78*x^3 - 6*x^4)*Log [x] + (105*x + 36*x^2 + 3*x^3)*Log[x]^2 + (-15*x^2 - 3*x^3 + (15*x + 3*x^2 )*Log[x])*Log[5 + x]),x]
Time = 4.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {7292, 7293, 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.
\(\displaystyle \int \frac {3 x^4+15 x^3+26 x^2+\left (3 x^2+15 x\right ) \log ^2(x)+\left (3 x^2+12 x-15\right ) \log (x+5)+\left (-6 x^3-36 x^2-27 x\right ) \log (x)+122 x-115}{3 x^5+42 x^4+112 x^3-115 x^2+\left (3 x^3+36 x^2+105 x\right ) \log ^2(x)+\left (-3 x^3-15 x^2+\left (3 x^2+15 x\right ) \log (x)\right ) \log (x+5)+\left (-6 x^4-78 x^3-217 x^2+115 x\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-3 x^4-15 x^3-26 x^2-\left (3 x^2+15 x\right ) \log ^2(x)-\left (3 x^2+12 x-15\right ) \log (x+5)-\left (-6 x^3-36 x^2-27 x\right ) \log (x)-122 x+115}{x (x+5) (x-\log (x)) \left (-3 x^2-27 x+3 x \log (x)+21 \log (x)+3 \log (x+5)+23\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 \left (2 x^3+18 x^2-x^2 \log (x)+32 x-5 x \log (x)-35\right )}{x (x+5) \left (3 x^2+27 x-3 x \log (x)-21 \log (x)-3 \log (x+5)-23\right )}+\frac {1-x}{x (x-\log (x))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (-3 x^2-27 x+3 x \log (x)+21 \log (x)+3 \log (x+5)+23\right )-\log (x-\log (x))\) |
Int[(-115 + 122*x + 26*x^2 + 15*x^3 + 3*x^4 + (-27*x - 36*x^2 - 6*x^3)*Log [x] + (15*x + 3*x^2)*Log[x]^2 + (-15 + 12*x + 3*x^2)*Log[5 + x])/(-115*x^2 + 112*x^3 + 42*x^4 + 3*x^5 + (115*x - 217*x^2 - 78*x^3 - 6*x^4)*Log[x] + (105*x + 36*x^2 + 3*x^3)*Log[x]^2 + (-15*x^2 - 3*x^3 + (15*x + 3*x^2)*Log[ x])*Log[5 + x]),x]
3.23.35.3.1 Defintions of rubi rules used
Time = 1.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36
method | result | size |
default | \(-\ln \left (x -\ln \left (x \right )\right )+\ln \left (x \ln \left (x \right )-x^{2}+7 \ln \left (x \right )-9 x +\ln \left (5+x \right )+\frac {23}{3}\right )\) | \(34\) |
risch | \(-\ln \left (\ln \left (x \right )-x \right )+\ln \left (x \ln \left (x \right )-x^{2}+7 \ln \left (x \right )-9 x +\ln \left (5+x \right )+\frac {23}{3}\right )\) | \(34\) |
parallelrisch | \(-\ln \left (x -\ln \left (x \right )\right )+\ln \left (x^{2}-x \ln \left (x \right )+9 x -7 \ln \left (x \right )-\ln \left (5+x \right )-\frac {23}{3}\right )\) | \(35\) |
int(((3*x^2+12*x-15)*ln(5+x)+(3*x^2+15*x)*ln(x)^2+(-6*x^3-36*x^2-27*x)*ln( x)+3*x^4+15*x^3+26*x^2+122*x-115)/(((3*x^2+15*x)*ln(x)-3*x^3-15*x^2)*ln(5+ x)+(3*x^3+36*x^2+105*x)*ln(x)^2+(-6*x^4-78*x^3-217*x^2+115*x)*ln(x)+3*x^5+ 42*x^4+112*x^3-115*x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-115+122 x+26 x^2+15 x^3+3 x^4+\left (-27 x-36 x^2-6 x^3\right ) \log (x)+\left (15 x+3 x^2\right ) \log ^2(x)+\left (-15+12 x+3 x^2\right ) \log (5+x)}{-115 x^2+112 x^3+42 x^4+3 x^5+\left (115 x-217 x^2-78 x^3-6 x^4\right ) \log (x)+\left (105 x+36 x^2+3 x^3\right ) \log ^2(x)+\left (-15 x^2-3 x^3+\left (15 x+3 x^2\right ) \log (x)\right ) \log (5+x)} \, dx=\log \left (-3 \, x^{2} + 3 \, {\left (x + 7\right )} \log \left (x\right ) - 27 \, x + 3 \, \log \left (x + 5\right ) + 23\right ) - \log \left (-x + \log \left (x\right )\right ) \]
integrate(((3*x^2+12*x-15)*log(5+x)+(3*x^2+15*x)*log(x)^2+(-6*x^3-36*x^2-2 7*x)*log(x)+3*x^4+15*x^3+26*x^2+122*x-115)/(((3*x^2+15*x)*log(x)-3*x^3-15* x^2)*log(5+x)+(3*x^3+36*x^2+105*x)*log(x)^2+(-6*x^4-78*x^3-217*x^2+115*x)* log(x)+3*x^5+42*x^4+112*x^3-115*x^2),x, algorithm=\
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {-115+122 x+26 x^2+15 x^3+3 x^4+\left (-27 x-36 x^2-6 x^3\right ) \log (x)+\left (15 x+3 x^2\right ) \log ^2(x)+\left (-15+12 x+3 x^2\right ) \log (5+x)}{-115 x^2+112 x^3+42 x^4+3 x^5+\left (115 x-217 x^2-78 x^3-6 x^4\right ) \log (x)+\left (105 x+36 x^2+3 x^3\right ) \log ^2(x)+\left (-15 x^2-3 x^3+\left (15 x+3 x^2\right ) \log (x)\right ) \log (5+x)} \, dx=- \log {\left (- x + \log {\left (x \right )} \right )} + \log {\left (- x^{2} + x \log {\left (x \right )} - 9 x + 7 \log {\left (x \right )} + \log {\left (x + 5 \right )} + \frac {23}{3} \right )} \]
integrate(((3*x**2+12*x-15)*ln(5+x)+(3*x**2+15*x)*ln(x)**2+(-6*x**3-36*x** 2-27*x)*ln(x)+3*x**4+15*x**3+26*x**2+122*x-115)/(((3*x**2+15*x)*ln(x)-3*x* *3-15*x**2)*ln(5+x)+(3*x**3+36*x**2+105*x)*ln(x)**2+(-6*x**4-78*x**3-217*x **2+115*x)*ln(x)+3*x**5+42*x**4+112*x**3-115*x**2),x)
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-115+122 x+26 x^2+15 x^3+3 x^4+\left (-27 x-36 x^2-6 x^3\right ) \log (x)+\left (15 x+3 x^2\right ) \log ^2(x)+\left (-15+12 x+3 x^2\right ) \log (5+x)}{-115 x^2+112 x^3+42 x^4+3 x^5+\left (115 x-217 x^2-78 x^3-6 x^4\right ) \log (x)+\left (105 x+36 x^2+3 x^3\right ) \log ^2(x)+\left (-15 x^2-3 x^3+\left (15 x+3 x^2\right ) \log (x)\right ) \log (5+x)} \, dx=\log \left (-x^{2} + {\left (x + 7\right )} \log \left (x\right ) - 9 \, x + \log \left (x + 5\right ) + \frac {23}{3}\right ) - \log \left (-x + \log \left (x\right )\right ) \]
integrate(((3*x^2+12*x-15)*log(5+x)+(3*x^2+15*x)*log(x)^2+(-6*x^3-36*x^2-2 7*x)*log(x)+3*x^4+15*x^3+26*x^2+122*x-115)/(((3*x^2+15*x)*log(x)-3*x^3-15* x^2)*log(5+x)+(3*x^3+36*x^2+105*x)*log(x)^2+(-6*x^4-78*x^3-217*x^2+115*x)* log(x)+3*x^5+42*x^4+112*x^3-115*x^2),x, algorithm=\
Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-115+122 x+26 x^2+15 x^3+3 x^4+\left (-27 x-36 x^2-6 x^3\right ) \log (x)+\left (15 x+3 x^2\right ) \log ^2(x)+\left (-15+12 x+3 x^2\right ) \log (5+x)}{-115 x^2+112 x^3+42 x^4+3 x^5+\left (115 x-217 x^2-78 x^3-6 x^4\right ) \log (x)+\left (105 x+36 x^2+3 x^3\right ) \log ^2(x)+\left (-15 x^2-3 x^3+\left (15 x+3 x^2\right ) \log (x)\right ) \log (5+x)} \, dx=\log \left (-3 \, x^{2} + 3 \, x \log \left (x\right ) - 27 \, x + 3 \, \log \left (x + 5\right ) + 21 \, \log \left (x\right ) + 23\right ) - \log \left (x - \log \left (x\right )\right ) \]
integrate(((3*x^2+12*x-15)*log(5+x)+(3*x^2+15*x)*log(x)^2+(-6*x^3-36*x^2-2 7*x)*log(x)+3*x^4+15*x^3+26*x^2+122*x-115)/(((3*x^2+15*x)*log(x)-3*x^3-15* x^2)*log(5+x)+(3*x^3+36*x^2+105*x)*log(x)^2+(-6*x^4-78*x^3-217*x^2+115*x)* log(x)+3*x^5+42*x^4+112*x^3-115*x^2),x, algorithm=\
Time = 11.83 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {-115+122 x+26 x^2+15 x^3+3 x^4+\left (-27 x-36 x^2-6 x^3\right ) \log (x)+\left (15 x+3 x^2\right ) \log ^2(x)+\left (-15+12 x+3 x^2\right ) \log (5+x)}{-115 x^2+112 x^3+42 x^4+3 x^5+\left (115 x-217 x^2-78 x^3-6 x^4\right ) \log (x)+\left (105 x+36 x^2+3 x^3\right ) \log ^2(x)+\left (-15 x^2-3 x^3+\left (15 x+3 x^2\right ) \log (x)\right ) \log (5+x)} \, dx=\ln \left (\ln \left (x+5\right )-9\,x+7\,\ln \left (x\right )+x\,\ln \left (x\right )-x^2+\frac {23}{3}\right )-\ln \left (\ln \left (x\right )-x\right ) \]
int((122*x + log(x + 5)*(12*x + 3*x^2 - 15) + log(x)^2*(15*x + 3*x^2) + 26 *x^2 + 15*x^3 + 3*x^4 - log(x)*(27*x + 36*x^2 + 6*x^3) - 115)/(log(x)^2*(1 05*x + 36*x^2 + 3*x^3) - log(x + 5)*(15*x^2 - log(x)*(15*x + 3*x^2) + 3*x^ 3) - log(x)*(217*x^2 - 115*x + 78*x^3 + 6*x^4) - 115*x^2 + 112*x^3 + 42*x^ 4 + 3*x^5),x)