Integrand size = 57, antiderivative size = 32 \[ \int \frac {175-45 x-663 x^2+61 x^3+54 x^4-8 x^5+\left (-175+70 x-7 x^2\right ) \log (x)}{50 x^2-20 x^3+2 x^4} \, dx=4+x-\frac {x}{5-x}+(14+2 x) \left (-x+\frac {\log (x)}{4 x}\right ) \] Output:
4+x+(14+2*x)*(1/4*ln(x)/x-x)-x/(5-x)
Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {175-45 x-663 x^2+61 x^3+54 x^4-8 x^5+\left (-175+70 x-7 x^2\right ) \log (x)}{50 x^2-20 x^3+2 x^4} \, dx=\frac {1}{2} \left (-\frac {10}{5-x}-26 x-4 x^2+\log (x)+\frac {7 \log (x)}{x}\right ) \] Input:
Integrate[(175 - 45*x - 663*x^2 + 61*x^3 + 54*x^4 - 8*x^5 + (-175 + 70*x - 7*x^2)*Log[x])/(50*x^2 - 20*x^3 + 2*x^4),x]
Output:
(-10/(5 - x) - 26*x - 4*x^2 + Log[x] + (7*Log[x])/x)/2
Time = 0.73 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2026, 7277, 27, 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.
\(\displaystyle \int \frac {-8 x^5+54 x^4+61 x^3-663 x^2+\left (-7 x^2+70 x-175\right ) \log (x)-45 x+175}{2 x^4-20 x^3+50 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-8 x^5+54 x^4+61 x^3-663 x^2+\left (-7 x^2+70 x-175\right ) \log (x)-45 x+175}{x^2 \left (2 x^2-20 x+50\right )}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 8 \int \frac {-8 x^5+54 x^4+61 x^3-663 x^2-45 x-7 \left (x^2-10 x+25\right ) \log (x)+175}{16 (5-x)^2 x^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {-8 x^5+54 x^4+61 x^3-663 x^2-45 x-7 \left (x^2-10 x+25\right ) \log (x)+175}{(5-x)^2 x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {8 x^3}{(x-5)^2}+\frac {54 x^2}{(x-5)^2}+\frac {61 x}{(x-5)^2}-\frac {663}{(x-5)^2}-\frac {45}{(x-5)^2 x}-\frac {7 \log (x)}{x^2}+\frac {175}{(x-5)^2 x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-4 x^2-26 x-\frac {10}{5-x}+\log (x)+\frac {7 \log (x)}{x}\right )\) |
Input:
Int[(175 - 45*x - 663*x^2 + 61*x^3 + 54*x^4 - 8*x^5 + (-175 + 70*x - 7*x^2 )*Log[x])/(50*x^2 - 20*x^3 + 2*x^4),x]
Output:
(-10/(5 - x) - 26*x - 4*x^2 + Log[x] + (7*Log[x])/x)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 0.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {7 \ln \left (x \right )}{2 x}-2 x^{2}-13 x +\frac {5}{-5+x}+\frac {\ln \left (x \right )}{2}\) | \(28\) |
parts | \(\frac {7 \ln \left (x \right )}{2 x}-2 x^{2}-13 x +\frac {5}{-5+x}+\frac {\ln \left (x \right )}{2}\) | \(28\) |
norman | \(\frac {330 x +x \ln \left (x \right )+\frac {x^{2} \ln \left (x \right )}{2}-3 x^{3}-2 x^{4}-\frac {35 \ln \left (x \right )}{2}}{x \left (-5+x \right )}\) | \(39\) |
risch | \(\frac {7 \ln \left (x \right )}{2 x}+\frac {-4 x^{3}+x \ln \left (x \right )-6 x^{2}-5 \ln \left (x \right )+130 x +10}{2 x -10}\) | \(39\) |
parallelrisch | \(\frac {-4 x^{4}-6 x^{3}+x^{2} \ln \left (x \right )+2 x \ln \left (x \right )+660 x -35 \ln \left (x \right )}{2 x \left (-5+x \right )}\) | \(40\) |
orering | \(-\frac {x \left (8 x^{7}-120 x^{6}+237 x^{5}+2685 x^{4}-978 x^{3}-88730 x^{2}+232025 x -18375\right ) \left (\left (-7 x^{2}+70 x -175\right ) \ln \left (x \right )-8 x^{5}+54 x^{4}+61 x^{3}-663 x^{2}-45 x +175\right )}{7 \left (24 x^{6}-308 x^{5}+1019 x^{4}+922 x^{3}-6780 x^{2}+650 x -875\right ) \left (2 x^{4}-20 x^{3}+50 x^{2}\right )}+\frac {\left (8 x^{6}+30 x^{5}-537 x^{4}+326 x^{2}+22590 x -1225\right ) x^{2} \left (-5+x \right ) \left (\frac {\left (-14 x +70\right ) \ln \left (x \right )+\frac {-7 x^{2}+70 x -175}{x}-40 x^{4}+216 x^{3}+183 x^{2}-1326 x -45}{2 x^{4}-20 x^{3}+50 x^{2}}-\frac {\left (\left (-7 x^{2}+70 x -175\right ) \ln \left (x \right )-8 x^{5}+54 x^{4}+61 x^{3}-663 x^{2}-45 x +175\right ) \left (8 x^{3}-60 x^{2}+100 x \right )}{\left (2 x^{4}-20 x^{3}+50 x^{2}\right )^{2}}\right )}{168 x^{6}-2156 x^{5}+7133 x^{4}+6454 x^{3}-47460 x^{2}+4550 x -6125}\) | \(327\) |
Input:
int(((-7*x^2+70*x-175)*ln(x)-8*x^5+54*x^4+61*x^3-663*x^2-45*x+175)/(2*x^4- 20*x^3+50*x^2),x,method=_RETURNVERBOSE)
Output:
7/2*ln(x)/x-2*x^2-13*x+5/(-5+x)+1/2*ln(x)
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {175-45 x-663 x^2+61 x^3+54 x^4-8 x^5+\left (-175+70 x-7 x^2\right ) \log (x)}{50 x^2-20 x^3+2 x^4} \, dx=-\frac {4 \, x^{4} + 6 \, x^{3} - 130 \, x^{2} - {\left (x^{2} + 2 \, x - 35\right )} \log \left (x\right ) - 10 \, x}{2 \, {\left (x^{2} - 5 \, x\right )}} \] Input:
integrate(((-7*x^2+70*x-175)*log(x)-8*x^5+54*x^4+61*x^3-663*x^2-45*x+175)/ (2*x^4-20*x^3+50*x^2),x, algorithm="fricas")
Output:
-1/2*(4*x^4 + 6*x^3 - 130*x^2 - (x^2 + 2*x - 35)*log(x) - 10*x)/(x^2 - 5*x )
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {175-45 x-663 x^2+61 x^3+54 x^4-8 x^5+\left (-175+70 x-7 x^2\right ) \log (x)}{50 x^2-20 x^3+2 x^4} \, dx=- 2 x^{2} - 13 x + \frac {\log {\left (x \right )}}{2} + \frac {5}{x - 5} + \frac {7 \log {\left (x \right )}}{2 x} \] Input:
integrate(((-7*x**2+70*x-175)*ln(x)-8*x**5+54*x**4+61*x**3-663*x**2-45*x+1 75)/(2*x**4-20*x**3+50*x**2),x)
Output:
-2*x**2 - 13*x + log(x)/2 + 5/(x - 5) + 7*log(x)/(2*x)
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {175-45 x-663 x^2+61 x^3+54 x^4-8 x^5+\left (-175+70 x-7 x^2\right ) \log (x)}{50 x^2-20 x^3+2 x^4} \, dx=-2 \, x^{2} - 13 \, x - \frac {7 \, {\left (2 \, x - 5\right )}}{2 \, {\left (x^{2} - 5 \, x\right )}} + \frac {7 \, {\left (\log \left (x\right ) + 1\right )}}{2 \, x} + \frac {17}{2 \, {\left (x - 5\right )}} + \frac {1}{2} \, \log \left (x\right ) \] Input:
integrate(((-7*x^2+70*x-175)*log(x)-8*x^5+54*x^4+61*x^3-663*x^2-45*x+175)/ (2*x^4-20*x^3+50*x^2),x, algorithm="maxima")
Output:
-2*x^2 - 13*x - 7/2*(2*x - 5)/(x^2 - 5*x) + 7/2*(log(x) + 1)/x + 17/2/(x - 5) + 1/2*log(x)
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {175-45 x-663 x^2+61 x^3+54 x^4-8 x^5+\left (-175+70 x-7 x^2\right ) \log (x)}{50 x^2-20 x^3+2 x^4} \, dx=-2 \, x^{2} - 13 \, x + \frac {7 \, \log \left (x\right )}{2 \, x} + \frac {5}{x - 5} + \frac {1}{2} \, \log \left (x\right ) \] Input:
integrate(((-7*x^2+70*x-175)*log(x)-8*x^5+54*x^4+61*x^3-663*x^2-45*x+175)/ (2*x^4-20*x^3+50*x^2),x, algorithm="giac")
Output:
-2*x^2 - 13*x + 7/2*log(x)/x + 5/(x - 5) + 1/2*log(x)
Time = 3.77 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {175-45 x-663 x^2+61 x^3+54 x^4-8 x^5+\left (-175+70 x-7 x^2\right ) \log (x)}{50 x^2-20 x^3+2 x^4} \, dx=\frac {\ln \left (x\right )}{2}-13\,x-2\,x^2-\frac {\frac {35\,\ln \left (x\right )}{2}-x\,\left (\frac {7\,\ln \left (x\right )}{2}+5\right )}{x\,\left (x-5\right )} \] Input:
int(-(45*x + log(x)*(7*x^2 - 70*x + 175) + 663*x^2 - 61*x^3 - 54*x^4 + 8*x ^5 - 175)/(50*x^2 - 20*x^3 + 2*x^4),x)
Output:
log(x)/2 - 13*x - 2*x^2 - ((35*log(x))/2 - x*((7*log(x))/2 + 5))/(x*(x - 5 ))
Time = 0.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {175-45 x-663 x^2+61 x^3+54 x^4-8 x^5+\left (-175+70 x-7 x^2\right ) \log (x)}{50 x^2-20 x^3+2 x^4} \, dx=\frac {5 \,\mathrm {log}\left (x \right ) x^{2}+10 \,\mathrm {log}\left (x \right ) x -175 \,\mathrm {log}\left (x \right )-20 x^{4}-30 x^{3}+653 x^{2}+35 x}{10 x \left (-5+x \right )} \] Input:
int(((-7*x^2+70*x-175)*log(x)-8*x^5+54*x^4+61*x^3-663*x^2-45*x+175)/(2*x^4 -20*x^3+50*x^2),x)
Output:
(5*log(x)*x**2 + 10*log(x)*x - 175*log(x) - 20*x**4 - 30*x**3 + 653*x**2 + 35*x)/(10*x*(x - 5))