Integrand size = 111, antiderivative size = 27 \[ \int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{2 x^2-4 x \log (x)+2 \log ^2(x)} \, dx=-1+x \left (\frac {3}{2}+x-\frac {x \left (5+x^2\right )^4}{-x+\log (x)}\right ) \] Output:
x*(x-(x^2+5)^4*x/(ln(x)-x)+3/2)-1
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{2 x^2-4 x \log (x)+2 \log ^2(x)} \, dx=\frac {1}{2} x \left (3+2 x+\frac {2 x \left (5+x^2\right )^4}{x-\log (x)}\right ) \] Input:
Integrate[(1250*x + 1253*x^2 + 1004*x^3 + 3000*x^4 + 300*x^5 + 1500*x^6 + 40*x^7 + 280*x^8 + 2*x^9 + 18*x^10 + (-2506*x - 8*x^2 - 4000*x^3 - 1800*x^ 5 - 320*x^7 - 20*x^9)*Log[x] + (3 + 4*x)*Log[x]^2)/(2*x^2 - 4*x*Log[x] + 2 *Log[x]^2),x]
Output:
(x*(3 + 2*x + (2*x*(5 + x^2)^4)/(x - Log[x])))/2
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.
\(\displaystyle \int \frac {18 x^{10}+2 x^9+280 x^8+40 x^7+1500 x^6+300 x^5+3000 x^4+1004 x^3+1253 x^2+\left (-20 x^9-320 x^7-1800 x^5-4000 x^3-8 x^2-2506 x\right ) \log (x)+1250 x+(4 x+3) \log ^2(x)}{2 x^2+2 \log ^2(x)-4 x \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {18 x^{10}+2 x^9+280 x^8+40 x^7+1500 x^6+300 x^5+3000 x^4+1004 x^3+1253 x^2+\left (-20 x^9-320 x^7-1800 x^5-4000 x^3-8 x^2-2506 x\right ) \log (x)+1250 x+(4 x+3) \log ^2(x)}{2 (x-\log (x))^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {18 x^{10}+2 x^9+280 x^8+40 x^7+1500 x^6+300 x^5+3000 x^4+1004 x^3+1253 x^2+1250 x+(4 x+3) \log ^2(x)-2 \left (10 x^9+160 x^7+900 x^5+2000 x^3+4 x^2+1253 x\right ) \log (x)}{(x-\log (x))^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {2 (x-1) x \left (x^2+5\right )^4}{(x-\log (x))^2}+\frac {20 x \left (x^2+1\right ) \left (x^2+5\right )^3}{x-\log (x)}+4 x+3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-2 \int \frac {x^{10}}{(x-\log (x))^2}dx+2 \int \frac {x^9}{(x-\log (x))^2}dx+20 \int \frac {x^9}{x-\log (x)}dx-40 \int \frac {x^8}{(x-\log (x))^2}dx+40 \int \frac {x^7}{(x-\log (x))^2}dx+320 \int \frac {x^7}{x-\log (x)}dx-300 \int \frac {x^6}{(x-\log (x))^2}dx+300 \int \frac {x^5}{(x-\log (x))^2}dx+1800 \int \frac {x^5}{x-\log (x)}dx-1000 \int \frac {x^4}{(x-\log (x))^2}dx+1000 \int \frac {x^3}{(x-\log (x))^2}dx+4000 \int \frac {x^3}{x-\log (x)}dx-1250 \int \frac {x^2}{(x-\log (x))^2}dx+1250 \int \frac {x}{(x-\log (x))^2}dx+2500 \int \frac {x}{x-\log (x)}dx+2 x^2+3 x\right )\) |
Input:
Int[(1250*x + 1253*x^2 + 1004*x^3 + 3000*x^4 + 300*x^5 + 1500*x^6 + 40*x^7 + 280*x^8 + 2*x^9 + 18*x^10 + (-2506*x - 8*x^2 - 4000*x^3 - 1800*x^5 - 32 0*x^7 - 20*x^9)*Log[x] + (3 + 4*x)*Log[x]^2)/(2*x^2 - 4*x*Log[x] + 2*Log[x ]^2),x]
Output:
$Aborted
Time = 2.56 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48
method | result | size |
risch | \(x^{2}+\frac {3 x}{2}+\frac {\left (x^{8}+20 x^{6}+150 x^{4}+500 x^{2}+625\right ) x^{2}}{x -\ln \left (x \right )}\) | \(40\) |
default | \(\frac {-2 x^{2} \ln \left (x \right )-3 x \ln \left (x \right )+1253 x^{2}+1000 x^{4}+300 x^{6}+2 x^{3}+40 x^{8}+2 x^{10}}{2 x -2 \ln \left (x \right )}\) | \(54\) |
parallelrisch | \(\frac {-2 x^{2} \ln \left (x \right )-3 x \ln \left (x \right )+1253 x^{2}+1000 x^{4}+300 x^{6}+2 x^{3}+40 x^{8}+2 x^{10}}{2 x -2 \ln \left (x \right )}\) | \(54\) |
Input:
int(((3+4*x)*ln(x)^2+(-20*x^9-320*x^7-1800*x^5-4000*x^3-8*x^2-2506*x)*ln(x )+18*x^10+2*x^9+280*x^8+40*x^7+1500*x^6+300*x^5+3000*x^4+1004*x^3+1253*x^2 +1250*x)/(2*ln(x)^2-4*x*ln(x)+2*x^2),x,method=_RETURNVERBOSE)
Output:
x^2+3/2*x+(x^8+20*x^6+150*x^4+500*x^2+625)*x^2/(x-ln(x))
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{2 x^2-4 x \log (x)+2 \log ^2(x)} \, dx=\frac {2 \, x^{10} + 40 \, x^{8} + 300 \, x^{6} + 1000 \, x^{4} + 2 \, x^{3} + 1253 \, x^{2} - {\left (2 \, x^{2} + 3 \, x\right )} \log \left (x\right )}{2 \, {\left (x - \log \left (x\right )\right )}} \] Input:
integrate(((3+4*x)*log(x)^2+(-20*x^9-320*x^7-1800*x^5-4000*x^3-8*x^2-2506* x)*log(x)+18*x^10+2*x^9+280*x^8+40*x^7+1500*x^6+300*x^5+3000*x^4+1004*x^3+ 1253*x^2+1250*x)/(2*log(x)^2-4*x*log(x)+2*x^2),x, algorithm="fricas")
Output:
1/2*(2*x^10 + 40*x^8 + 300*x^6 + 1000*x^4 + 2*x^3 + 1253*x^2 - (2*x^2 + 3* x)*log(x))/(x - log(x))
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{2 x^2-4 x \log (x)+2 \log ^2(x)} \, dx=x^{2} + \frac {3 x}{2} + \frac {- x^{10} - 20 x^{8} - 150 x^{6} - 500 x^{4} - 625 x^{2}}{- x + \log {\left (x \right )}} \] Input:
integrate(((3+4*x)*ln(x)**2+(-20*x**9-320*x**7-1800*x**5-4000*x**3-8*x**2- 2506*x)*ln(x)+18*x**10+2*x**9+280*x**8+40*x**7+1500*x**6+300*x**5+3000*x** 4+1004*x**3+1253*x**2+1250*x)/(2*ln(x)**2-4*x*ln(x)+2*x**2),x)
Output:
x**2 + 3*x/2 + (-x**10 - 20*x**8 - 150*x**6 - 500*x**4 - 625*x**2)/(-x + l og(x))
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{2 x^2-4 x \log (x)+2 \log ^2(x)} \, dx=\frac {2 \, x^{10} + 40 \, x^{8} + 300 \, x^{6} + 1000 \, x^{4} + 2 \, x^{3} + 1253 \, x^{2} - {\left (2 \, x^{2} + 3 \, x\right )} \log \left (x\right )}{2 \, {\left (x - \log \left (x\right )\right )}} \] Input:
integrate(((3+4*x)*log(x)^2+(-20*x^9-320*x^7-1800*x^5-4000*x^3-8*x^2-2506* x)*log(x)+18*x^10+2*x^9+280*x^8+40*x^7+1500*x^6+300*x^5+3000*x^4+1004*x^3+ 1253*x^2+1250*x)/(2*log(x)^2-4*x*log(x)+2*x^2),x, algorithm="maxima")
Output:
1/2*(2*x^10 + 40*x^8 + 300*x^6 + 1000*x^4 + 2*x^3 + 1253*x^2 - (2*x^2 + 3* x)*log(x))/(x - log(x))
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{2 x^2-4 x \log (x)+2 \log ^2(x)} \, dx=x^{2} + \frac {3}{2} \, x + \frac {x^{10} + 20 \, x^{8} + 150 \, x^{6} + 500 \, x^{4} + 625 \, x^{2}}{x - \log \left (x\right )} \] Input:
integrate(((3+4*x)*log(x)^2+(-20*x^9-320*x^7-1800*x^5-4000*x^3-8*x^2-2506* x)*log(x)+18*x^10+2*x^9+280*x^8+40*x^7+1500*x^6+300*x^5+3000*x^4+1004*x^3+ 1253*x^2+1250*x)/(2*log(x)^2-4*x*log(x)+2*x^2),x, algorithm="giac")
Output:
x^2 + 3/2*x + (x^10 + 20*x^8 + 150*x^6 + 500*x^4 + 625*x^2)/(x - log(x))
Time = 2.89 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.26 \[ \int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{2 x^2-4 x \log (x)+2 \log ^2(x)} \, dx=\frac {x\,\left (2\,x+3\right )}{2}+\frac {\frac {x\,\left (2\,x^9+40\,x^7+300\,x^5+1000\,x^3+2\,x^2+1253\,x\right )}{2}-\frac {x^2\,\left (2\,x+3\right )}{2}}{x-\ln \left (x\right )} \] Input:
int((1250*x - log(x)*(2506*x + 8*x^2 + 4000*x^3 + 1800*x^5 + 320*x^7 + 20* x^9) + 1253*x^2 + 1004*x^3 + 3000*x^4 + 300*x^5 + 1500*x^6 + 40*x^7 + 280* x^8 + 2*x^9 + 18*x^10 + log(x)^2*(4*x + 3))/(2*log(x)^2 - 4*x*log(x) + 2*x ^2),x)
Output:
(x*(2*x + 3))/2 + ((x*(1253*x + 2*x^2 + 1000*x^3 + 300*x^5 + 40*x^7 + 2*x^ 9))/2 - (x^2*(2*x + 3))/2)/(x - log(x))
Time = 0.52 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85 \[ \int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{2 x^2-4 x \log (x)+2 \log ^2(x)} \, dx=\frac {x \left (2 \,\mathrm {log}\left (x \right ) x +3 \,\mathrm {log}\left (x \right )-2 x^{9}-40 x^{7}-300 x^{5}-1000 x^{3}-2 x^{2}-1253 x \right )}{2 \,\mathrm {log}\left (x \right )-2 x} \] Input:
int(((3+4*x)*log(x)^2+(-20*x^9-320*x^7-1800*x^5-4000*x^3-8*x^2-2506*x)*log (x)+18*x^10+2*x^9+280*x^8+40*x^7+1500*x^6+300*x^5+3000*x^4+1004*x^3+1253*x ^2+1250*x)/(2*log(x)^2-4*x*log(x)+2*x^2),x)
Output:
(x*(2*log(x)*x + 3*log(x) - 2*x**9 - 40*x**7 - 300*x**5 - 1000*x**3 - 2*x* *2 - 1253*x))/(2*(log(x) - x))