Integrand size = 82, antiderivative size = 20 \[ \int \frac {-20+19 x-3 x^2+\left (4 x+x^2\right ) \log \left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )}{\left (16 x+4 x^2\right ) \log ^2\left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )} \, dx=\frac {-5+x}{4 \log \left (\frac {5 (4+x)^4}{x}\right )} \] Output:
1/4*(-5+x)/ln(5/x*(4+x)^4)
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-20+19 x-3 x^2+\left (4 x+x^2\right ) \log \left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )}{\left (16 x+4 x^2\right ) \log ^2\left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )} \, dx=\frac {-5+x}{4 \log \left (\frac {5 (4+x)^4}{x}\right )} \] Input:
Integrate[(-20 + 19*x - 3*x^2 + (4*x + x^2)*Log[(1280 + 1280*x + 480*x^2 + 80*x^3 + 5*x^4)/x])/((16*x + 4*x^2)*Log[(1280 + 1280*x + 480*x^2 + 80*x^3 + 5*x^4)/x]^2),x]
Output:
(-5 + x)/(4*Log[(5*(4 + x)^4)/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.
\(\displaystyle \int \frac {-3 x^2+\left (x^2+4 x\right ) \log \left (\frac {5 x^4+80 x^3+480 x^2+1280 x+1280}{x}\right )+19 x-20}{\left (4 x^2+16 x\right ) \log ^2\left (\frac {5 x^4+80 x^3+480 x^2+1280 x+1280}{x}\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-3 x^2+\left (x^2+4 x\right ) \log \left (\frac {5 x^4+80 x^3+480 x^2+1280 x+1280}{x}\right )+19 x-20}{x (4 x+16) \log ^2\left (\frac {5 x^4+80 x^3+480 x^2+1280 x+1280}{x}\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-3 x^2+\left (x^2+4 x\right ) \log \left (\frac {5 x^4+80 x^3+480 x^2+1280 x+1280}{x}\right )+19 x-20}{x (4 x+16) \log ^2\left (\frac {5 (x+4)^4}{x}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-3 x^2+19 x-20}{4 x (x+4) \log ^2\left (\frac {5 (x+4)^4}{x}\right )}+\frac {1}{4 \log \left (\frac {5 (x+4)^4}{x}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \int \frac {-3 x^2+19 x-20}{x (x+4) \log ^2\left (\frac {5 (x+4)^4}{x}\right )}dx+\frac {1}{4} \int \frac {1}{\log \left (\frac {5 (x+4)^4}{x}\right )}dx\) |
Input:
Int[(-20 + 19*x - 3*x^2 + (4*x + x^2)*Log[(1280 + 1280*x + 480*x^2 + 80*x^ 3 + 5*x^4)/x])/((16*x + 4*x^2)*Log[(1280 + 1280*x + 480*x^2 + 80*x^3 + 5*x ^4)/x]^2),x]
Output:
$Aborted
Time = 2.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65
method | result | size |
risch | \(\frac {-5+x}{4 \ln \left (\frac {5 x^{4}+80 x^{3}+480 x^{2}+1280 x +1280}{x}\right )}\) | \(33\) |
norman | \(\frac {\frac {x}{4}-\frac {5}{4}}{\ln \left (\frac {5 x^{4}+80 x^{3}+480 x^{2}+1280 x +1280}{x}\right )}\) | \(34\) |
parallelrisch | \(-\frac {5-x}{4 \ln \left (\frac {5 x^{4}+80 x^{3}+480 x^{2}+1280 x +1280}{x}\right )}\) | \(34\) |
Input:
int(((x^2+4*x)*ln((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)-3*x^2+19*x-20)/(4* x^2+16*x)/ln((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)^2,x,method=_RETURNVERBO SE)
Output:
1/4*(-5+x)/ln((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {-20+19 x-3 x^2+\left (4 x+x^2\right ) \log \left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )}{\left (16 x+4 x^2\right ) \log ^2\left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )} \, dx=\frac {x - 5}{4 \, \log \left (\frac {5 \, {\left (x^{4} + 16 \, x^{3} + 96 \, x^{2} + 256 \, x + 256\right )}}{x}\right )} \] Input:
integrate(((x^2+4*x)*log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)-3*x^2+19*x- 20)/(4*x^2+16*x)/log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)^2,x, algorithm= "fricas")
Output:
1/4*(x - 5)/log(5*(x^4 + 16*x^3 + 96*x^2 + 256*x + 256)/x)
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {-20+19 x-3 x^2+\left (4 x+x^2\right ) \log \left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )}{\left (16 x+4 x^2\right ) \log ^2\left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )} \, dx=\frac {x - 5}{4 \log {\left (\frac {5 x^{4} + 80 x^{3} + 480 x^{2} + 1280 x + 1280}{x} \right )}} \] Input:
integrate(((x**2+4*x)*ln((5*x**4+80*x**3+480*x**2+1280*x+1280)/x)-3*x**2+1 9*x-20)/(4*x**2+16*x)/ln((5*x**4+80*x**3+480*x**2+1280*x+1280)/x)**2,x)
Output:
(x - 5)/(4*log((5*x**4 + 80*x**3 + 480*x**2 + 1280*x + 1280)/x))
Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-20+19 x-3 x^2+\left (4 x+x^2\right ) \log \left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )}{\left (16 x+4 x^2\right ) \log ^2\left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )} \, dx=\frac {x - 5}{4 \, {\left (\log \left (5\right ) + 4 \, \log \left (x + 4\right ) - \log \left (x\right )\right )}} \] Input:
integrate(((x^2+4*x)*log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)-3*x^2+19*x- 20)/(4*x^2+16*x)/log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)^2,x, algorithm= "maxima")
Output:
1/4*(x - 5)/(log(5) + 4*log(x + 4) - log(x))
Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {-20+19 x-3 x^2+\left (4 x+x^2\right ) \log \left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )}{\left (16 x+4 x^2\right ) \log ^2\left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )} \, dx=\frac {x - 5}{4 \, \log \left (\frac {5 \, {\left (x^{4} + 16 \, x^{3} + 96 \, x^{2} + 256 \, x + 256\right )}}{x}\right )} \] Input:
integrate(((x^2+4*x)*log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)-3*x^2+19*x- 20)/(4*x^2+16*x)/log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)^2,x, algorithm= "giac")
Output:
1/4*(x - 5)/log(5*(x^4 + 16*x^3 + 96*x^2 + 256*x + 256)/x)
Time = 2.87 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {-20+19 x-3 x^2+\left (4 x+x^2\right ) \log \left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )}{\left (16 x+4 x^2\right ) \log ^2\left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )} \, dx=\frac {x-5}{4\,\ln \left (\frac {5\,x^4+80\,x^3+480\,x^2+1280\,x+1280}{x}\right )} \] Input:
int((19*x - 3*x^2 + log((1280*x + 480*x^2 + 80*x^3 + 5*x^4 + 1280)/x)*(4*x + x^2) - 20)/(log((1280*x + 480*x^2 + 80*x^3 + 5*x^4 + 1280)/x)^2*(16*x + 4*x^2)),x)
Output:
(x - 5)/(4*log((1280*x + 480*x^2 + 80*x^3 + 5*x^4 + 1280)/x))
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {-20+19 x-3 x^2+\left (4 x+x^2\right ) \log \left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )}{\left (16 x+4 x^2\right ) \log ^2\left (\frac {1280+1280 x+480 x^2+80 x^3+5 x^4}{x}\right )} \, dx=\frac {-5+x}{4 \,\mathrm {log}\left (\frac {5 x^{4}+80 x^{3}+480 x^{2}+1280 x +1280}{x}\right )} \] Input:
int(((x^2+4*x)*log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)-3*x^2+19*x-20)/(4 *x^2+16*x)/log((5*x^4+80*x^3+480*x^2+1280*x+1280)/x)^2,x)
Output:
(x - 5)/(4*log((5*x**4 + 80*x**3 + 480*x**2 + 1280*x + 1280)/x))