Integrand size = 124, antiderivative size = 20 \[ \int \frac {-90 x^2-18 x^3-2 x^4-2 x^3 \log (x)+\left (-10 x^2+18 x^3+4 x^4+\left (30 x^2+6 x^3\right ) \log (x)\right ) \log (5+x)}{45+9 x+\left (-30 x-6 x^2+(-30-6 x) \log (x)\right ) \log (5+x)+\left (5 x^2+x^3+\left (10 x+2 x^2\right ) \log (x)+(5+x) \log ^2(x)\right ) \log ^2(5+x)} \, dx=27+\frac {2 x^3}{-3+(x+\log (x)) \log (5+x)} \] Output:
27+2*x^3/(ln(5+x)*(x+ln(x))-3)
Time = 0.43 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-90 x^2-18 x^3-2 x^4-2 x^3 \log (x)+\left (-10 x^2+18 x^3+4 x^4+\left (30 x^2+6 x^3\right ) \log (x)\right ) \log (5+x)}{45+9 x+\left (-30 x-6 x^2+(-30-6 x) \log (x)\right ) \log (5+x)+\left (5 x^2+x^3+\left (10 x+2 x^2\right ) \log (x)+(5+x) \log ^2(x)\right ) \log ^2(5+x)} \, dx=\frac {2 x^3}{-3+(x+\log (x)) \log (5+x)} \] Input:
Integrate[(-90*x^2 - 18*x^3 - 2*x^4 - 2*x^3*Log[x] + (-10*x^2 + 18*x^3 + 4 *x^4 + (30*x^2 + 6*x^3)*Log[x])*Log[5 + x])/(45 + 9*x + (-30*x - 6*x^2 + ( -30 - 6*x)*Log[x])*Log[5 + x] + (5*x^2 + x^3 + (10*x + 2*x^2)*Log[x] + (5 + x)*Log[x]^2)*Log[5 + x]^2),x]
Output:
(2*x^3)/(-3 + (x + Log[x])*Log[5 + 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.
| \(\displaystyle \int \frac {-2 x^4-18 x^3-2 x^3 \log (x)-90 x^2+\left (4 x^4+18 x^3-10 x^2+\left (6 x^3+30 x^2\right ) \log (x)\right ) \log (x+5)}{\left (-6 x^2-30 x+(-6 x-30) \log (x)\right ) \log (x+5)+\left (x^3+5 x^2+\left (2 x^2+10 x\right ) \log (x)+(x+5) \log ^2(x)\right ) \log ^2(x+5)+9 x+45} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x^2 \left (-x^2+\left (2 x^2+9 x-5\right ) \log (x+5)-9 x+\log (x) (3 (x+5) \log (x+5)-x)-45\right )}{(x+5) (3-(x+\log (x)) \log (x+5))^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {x^2 \left (x^2+9 x+\left (-2 x^2-9 x+5\right ) \log (x+5)+\log (x) (x-3 (x+5) \log (x+5))+45\right )}{(x+5) (3-(x+\log (x)) \log (x+5))^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {x^2 \left (x^2+9 x+\left (-2 x^2-9 x+5\right ) \log (x+5)+\log (x) (x-3 (x+5) \log (x+5))+45\right )}{(x+5) (3-(x+\log (x)) \log (x+5))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {x^2 \left (x^3+2 \log (x) x^2+3 x^2+\log ^2(x) x+18 x+15\right )}{(x+5) (x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}-\frac {x^2 (2 x+3 \log (x)-1)}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\int \frac {x^4}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx-2 \int \frac {x^3}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx+2 \int \frac {x^3 \log (x)}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx-2 \int \frac {x^3}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)}dx+\int \frac {x^2 \log ^2(x)}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx+28 \int \frac {x^2}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx-10 \int \frac {x^2 \log (x)}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx+\int \frac {x^2}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)}dx-3 \int \frac {x^2 \log (x)}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)}dx+25 \int \frac {\log ^2(x)}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx-5 \int \frac {x \log ^2(x)}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx-125 \int \frac {\log ^2(x)}{(x+5) (x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx+625 \int \frac {1}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx-125 \int \frac {x}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx-3125 \int \frac {1}{(x+5) (x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx-250 \int \frac {\log (x)}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx+50 \int \frac {x \log (x)}{(x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx+1250 \int \frac {\log (x)}{(x+5) (x+\log (x)) (x \log (x+5)+\log (x) \log (x+5)-3)^2}dx\right )\) |
Input:
Int[(-90*x^2 - 18*x^3 - 2*x^4 - 2*x^3*Log[x] + (-10*x^2 + 18*x^3 + 4*x^4 + (30*x^2 + 6*x^3)*Log[x])*Log[5 + x])/(45 + 9*x + (-30*x - 6*x^2 + (-30 - 6*x)*Log[x])*Log[5 + x] + (5*x^2 + x^3 + (10*x + 2*x^2)*Log[x] + (5 + x)*L og[x]^2)*Log[5 + x]^2),x]
Output:
$Aborted
Time = 2.46 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15
| method | result | size |
| default | \(\frac {2 x^{3}}{\ln \left (x \right ) \ln \left (5+x \right )+x \ln \left (5+x \right )-3}\) | \(23\) |
| risch | \(\frac {2 x^{3}}{\ln \left (x \right ) \ln \left (5+x \right )+x \ln \left (5+x \right )-3}\) | \(23\) |
| parallelrisch | \(\frac {2 x^{3}}{\ln \left (x \right ) \ln \left (5+x \right )+x \ln \left (5+x \right )-3}\) | \(23\) |
Input:
int((((6*x^3+30*x^2)*ln(x)+4*x^4+18*x^3-10*x^2)*ln(5+x)-2*x^3*ln(x)-2*x^4- 18*x^3-90*x^2)/(((5+x)*ln(x)^2+(2*x^2+10*x)*ln(x)+x^3+5*x^2)*ln(5+x)^2+((- 6*x-30)*ln(x)-6*x^2-30*x)*ln(5+x)+9*x+45),x,method=_RETURNVERBOSE)
Output:
2*x^3/(ln(x)*ln(5+x)+x*ln(5+x)-3)
Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-90 x^2-18 x^3-2 x^4-2 x^3 \log (x)+\left (-10 x^2+18 x^3+4 x^4+\left (30 x^2+6 x^3\right ) \log (x)\right ) \log (5+x)}{45+9 x+\left (-30 x-6 x^2+(-30-6 x) \log (x)\right ) \log (5+x)+\left (5 x^2+x^3+\left (10 x+2 x^2\right ) \log (x)+(5+x) \log ^2(x)\right ) \log ^2(5+x)} \, dx=\frac {2 \, x^{3}}{{\left (x + \log \left (x\right )\right )} \log \left (x + 5\right ) - 3} \] Input:
integrate((((6*x^3+30*x^2)*log(x)+4*x^4+18*x^3-10*x^2)*log(5+x)-2*x^3*log( x)-2*x^4-18*x^3-90*x^2)/(((5+x)*log(x)^2+(2*x^2+10*x)*log(x)+x^3+5*x^2)*lo g(5+x)^2+((-6*x-30)*log(x)-6*x^2-30*x)*log(5+x)+9*x+45),x, algorithm="fric as")
Output:
2*x^3/((x + log(x))*log(x + 5) - 3)
Time = 0.14 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {-90 x^2-18 x^3-2 x^4-2 x^3 \log (x)+\left (-10 x^2+18 x^3+4 x^4+\left (30 x^2+6 x^3\right ) \log (x)\right ) \log (5+x)}{45+9 x+\left (-30 x-6 x^2+(-30-6 x) \log (x)\right ) \log (5+x)+\left (5 x^2+x^3+\left (10 x+2 x^2\right ) \log (x)+(5+x) \log ^2(x)\right ) \log ^2(5+x)} \, dx=\frac {2 x^{3}}{\left (x + \log {\left (x \right )}\right ) \log {\left (x + 5 \right )} - 3} \] Input:
integrate((((6*x**3+30*x**2)*ln(x)+4*x**4+18*x**3-10*x**2)*ln(5+x)-2*x**3* ln(x)-2*x**4-18*x**3-90*x**2)/(((5+x)*ln(x)**2+(2*x**2+10*x)*ln(x)+x**3+5* x**2)*ln(5+x)**2+((-6*x-30)*ln(x)-6*x**2-30*x)*ln(5+x)+9*x+45),x)
Output:
2*x**3/((x + log(x))*log(x + 5) - 3)
Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-90 x^2-18 x^3-2 x^4-2 x^3 \log (x)+\left (-10 x^2+18 x^3+4 x^4+\left (30 x^2+6 x^3\right ) \log (x)\right ) \log (5+x)}{45+9 x+\left (-30 x-6 x^2+(-30-6 x) \log (x)\right ) \log (5+x)+\left (5 x^2+x^3+\left (10 x+2 x^2\right ) \log (x)+(5+x) \log ^2(x)\right ) \log ^2(5+x)} \, dx=\frac {2 \, x^{3}}{{\left (x + \log \left (x\right )\right )} \log \left (x + 5\right ) - 3} \] Input:
integrate((((6*x^3+30*x^2)*log(x)+4*x^4+18*x^3-10*x^2)*log(5+x)-2*x^3*log( x)-2*x^4-18*x^3-90*x^2)/(((5+x)*log(x)^2+(2*x^2+10*x)*log(x)+x^3+5*x^2)*lo g(5+x)^2+((-6*x-30)*log(x)-6*x^2-30*x)*log(5+x)+9*x+45),x, algorithm="maxi ma")
Output:
2*x^3/((x + log(x))*log(x + 5) - 3)
Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {-90 x^2-18 x^3-2 x^4-2 x^3 \log (x)+\left (-10 x^2+18 x^3+4 x^4+\left (30 x^2+6 x^3\right ) \log (x)\right ) \log (5+x)}{45+9 x+\left (-30 x-6 x^2+(-30-6 x) \log (x)\right ) \log (5+x)+\left (5 x^2+x^3+\left (10 x+2 x^2\right ) \log (x)+(5+x) \log ^2(x)\right ) \log ^2(5+x)} \, dx=\frac {2 \, x^{3}}{x \log \left (x + 5\right ) + \log \left (x + 5\right ) \log \left (x\right ) - 3} \] Input:
integrate((((6*x^3+30*x^2)*log(x)+4*x^4+18*x^3-10*x^2)*log(5+x)-2*x^3*log( x)-2*x^4-18*x^3-90*x^2)/(((5+x)*log(x)^2+(2*x^2+10*x)*log(x)+x^3+5*x^2)*lo g(5+x)^2+((-6*x-30)*log(x)-6*x^2-30*x)*log(5+x)+9*x+45),x, algorithm="giac ")
Output:
2*x^3/(x*log(x + 5) + log(x + 5)*log(x) - 3)
Timed out. \[ \int \frac {-90 x^2-18 x^3-2 x^4-2 x^3 \log (x)+\left (-10 x^2+18 x^3+4 x^4+\left (30 x^2+6 x^3\right ) \log (x)\right ) \log (5+x)}{45+9 x+\left (-30 x-6 x^2+(-30-6 x) \log (x)\right ) \log (5+x)+\left (5 x^2+x^3+\left (10 x+2 x^2\right ) \log (x)+(5+x) \log ^2(x)\right ) \log ^2(5+x)} \, dx=\int -\frac {2\,x^3\,\ln \left (x\right )-\ln \left (x+5\right )\,\left (\ln \left (x\right )\,\left (6\,x^3+30\,x^2\right )-10\,x^2+18\,x^3+4\,x^4\right )+90\,x^2+18\,x^3+2\,x^4}{\left ({\ln \left (x\right )}^2\,\left (x+5\right )+\ln \left (x\right )\,\left (2\,x^2+10\,x\right )+5\,x^2+x^3\right )\,{\ln \left (x+5\right )}^2+\left (-30\,x-\ln \left (x\right )\,\left (6\,x+30\right )-6\,x^2\right )\,\ln \left (x+5\right )+9\,x+45} \,d x \] Input:
int(-(2*x^3*log(x) - log(x + 5)*(log(x)*(30*x^2 + 6*x^3) - 10*x^2 + 18*x^3 + 4*x^4) + 90*x^2 + 18*x^3 + 2*x^4)/(9*x - log(x + 5)*(30*x + log(x)*(6*x + 30) + 6*x^2) + log(x + 5)^2*(log(x)^2*(x + 5) + log(x)*(10*x + 2*x^2) + 5*x^2 + x^3) + 45),x)
Output:
int(-(2*x^3*log(x) - log(x + 5)*(log(x)*(30*x^2 + 6*x^3) - 10*x^2 + 18*x^3 + 4*x^4) + 90*x^2 + 18*x^3 + 2*x^4)/(9*x - log(x + 5)*(30*x + log(x)*(6*x + 30) + 6*x^2) + log(x + 5)^2*(log(x)^2*(x + 5) + log(x)*(10*x + 2*x^2) + 5*x^2 + x^3) + 45), x)
Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {-90 x^2-18 x^3-2 x^4-2 x^3 \log (x)+\left (-10 x^2+18 x^3+4 x^4+\left (30 x^2+6 x^3\right ) \log (x)\right ) \log (5+x)}{45+9 x+\left (-30 x-6 x^2+(-30-6 x) \log (x)\right ) \log (5+x)+\left (5 x^2+x^3+\left (10 x+2 x^2\right ) \log (x)+(5+x) \log ^2(x)\right ) \log ^2(5+x)} \, dx=\frac {2 x^{3}}{\mathrm {log}\left (x +5\right ) \mathrm {log}\left (x \right )+\mathrm {log}\left (x +5\right ) x -3} \] Input:
int((((6*x^3+30*x^2)*log(x)+4*x^4+18*x^3-10*x^2)*log(5+x)-2*x^3*log(x)-2*x ^4-18*x^3-90*x^2)/(((5+x)*log(x)^2+(2*x^2+10*x)*log(x)+x^3+5*x^2)*log(5+x) ^2+((-6*x-30)*log(x)-6*x^2-30*x)*log(5+x)+9*x+45),x)
Output:
(2*x**3)/(log(x + 5)*log(x) + log(x + 5)*x - 3)