Integrand size = 92, antiderivative size = 34 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=1+\frac {1}{4} \log \left (x+x^2+\frac {\left (\frac {2+\frac {5}{x^2}-x}{x}+x\right )^2}{x^2}\right ) \] Output:
1/4*ln(x^2+((5/x^2+2-x)/x+x)^2/x^2+x)+1
Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {1}{4} \left (-8 \log (x)+\log \left (25+20 x^2-10 x^3+14 x^4-4 x^5+5 x^6-2 x^7+x^8+x^9+x^{10}\right )\right ) \] Input:
Integrate[(-200 - 120*x^2 + 50*x^3 - 56*x^4 + 12*x^5 - 10*x^6 + 2*x^7 + x^ 9 + 2*x^10)/(100*x + 80*x^3 - 40*x^4 + 56*x^5 - 16*x^6 + 20*x^7 - 8*x^8 + 4*x^9 + 4*x^10 + 4*x^11),x]
Output:
(-8*Log[x] + Log[25 + 20*x^2 - 10*x^3 + 14*x^4 - 4*x^5 + 5*x^6 - 2*x^7 + x ^8 + x^9 + x^10])/4
Time = 1.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2026, 7292, 27, 25, 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.
\(\displaystyle \int \frac {2 x^{10}+x^9+2 x^7-10 x^6+12 x^5-56 x^4+50 x^3-120 x^2-200}{4 x^{11}+4 x^{10}+4 x^9-8 x^8+20 x^7-16 x^6+56 x^5-40 x^4+80 x^3+100 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {2 x^{10}+x^9+2 x^7-10 x^6+12 x^5-56 x^4+50 x^3-120 x^2-200}{x \left (4 x^{10}+4 x^9+4 x^8-8 x^7+20 x^6-16 x^5+56 x^4-40 x^3+80 x^2+100\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^{10}+x^9+2 x^7-10 x^6+12 x^5-56 x^4+50 x^3-120 x^2-200}{4 x \left (x^{10}+x^9+x^8-2 x^7+5 x^6-4 x^5+14 x^4-10 x^3+20 x^2+25\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int -\frac {-2 x^{10}-x^9-2 x^7+10 x^6-12 x^5+56 x^4-50 x^3+120 x^2+200}{x \left (x^{10}+x^9+x^8-2 x^7+5 x^6-4 x^5+14 x^4-10 x^3+20 x^2+25\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{4} \int \frac {-2 x^{10}-x^9-2 x^7+10 x^6-12 x^5+56 x^4-50 x^3+120 x^2+200}{x \left (x^{10}+x^9+x^8-2 x^7+5 x^6-4 x^5+14 x^4-10 x^3+20 x^2+25\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{4} \int \left (\frac {8}{x}-\frac {x \left (10 x^8+9 x^7+8 x^6-14 x^5+30 x^4-20 x^3+56 x^2-30 x+40\right )}{x^{10}+x^9+x^8-2 x^7+5 x^6-4 x^5+14 x^4-10 x^3+20 x^2+25}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (\log \left (x^{10}+x^9+x^8-2 x^7+5 x^6-4 x^5+14 x^4-10 x^3+20 x^2+25\right )-8 \log (x)\right )\) |
Input:
Int[(-200 - 120*x^2 + 50*x^3 - 56*x^4 + 12*x^5 - 10*x^6 + 2*x^7 + x^9 + 2* x^10)/(100*x + 80*x^3 - 40*x^4 + 56*x^5 - 16*x^6 + 20*x^7 - 8*x^8 + 4*x^9 + 4*x^10 + 4*x^11),x]
Output:
(-8*Log[x] + Log[25 + 20*x^2 - 10*x^3 + 14*x^4 - 4*x^5 + 5*x^6 - 2*x^7 + x ^8 + x^9 + x^10])/4
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])
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47
method | result | size |
default | \(-2 \ln \left (x \right )+\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}\) | \(50\) |
norman | \(-2 \ln \left (x \right )+\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}\) | \(50\) |
risch | \(-2 \ln \left (x \right )+\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}\) | \(50\) |
parallelrisch | \(-2 \ln \left (x \right )+\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}\) | \(50\) |
Input:
int((2*x^10+x^9+2*x^7-10*x^6+12*x^5-56*x^4+50*x^3-120*x^2-200)/(4*x^11+4*x ^10+4*x^9-8*x^8+20*x^7-16*x^6+56*x^5-40*x^4+80*x^3+100*x),x,method=_RETURN VERBOSE)
Output:
-2*ln(x)+1/4*ln(x^10+x^9+x^8-2*x^7+5*x^6-4*x^5+14*x^4-10*x^3+20*x^2+25)
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {1}{4} \, \log \left (x^{10} + x^{9} + x^{8} - 2 \, x^{7} + 5 \, x^{6} - 4 \, x^{5} + 14 \, x^{4} - 10 \, x^{3} + 20 \, x^{2} + 25\right ) - 2 \, \log \left (x\right ) \] Input:
integrate((2*x^10+x^9+2*x^7-10*x^6+12*x^5-56*x^4+50*x^3-120*x^2-200)/(4*x^ 11+4*x^10+4*x^9-8*x^8+20*x^7-16*x^6+56*x^5-40*x^4+80*x^3+100*x),x, algorit hm="fricas")
Output:
1/4*log(x^10 + x^9 + x^8 - 2*x^7 + 5*x^6 - 4*x^5 + 14*x^4 - 10*x^3 + 20*x^ 2 + 25) - 2*log(x)
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=- 2 \log {\left (x \right )} + \frac {\log {\left (x^{10} + x^{9} + x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} + 14 x^{4} - 10 x^{3} + 20 x^{2} + 25 \right )}}{4} \] Input:
integrate((2*x**10+x**9+2*x**7-10*x**6+12*x**5-56*x**4+50*x**3-120*x**2-20 0)/(4*x**11+4*x**10+4*x**9-8*x**8+20*x**7-16*x**6+56*x**5-40*x**4+80*x**3+ 100*x),x)
Output:
-2*log(x) + log(x**10 + x**9 + x**8 - 2*x**7 + 5*x**6 - 4*x**5 + 14*x**4 - 10*x**3 + 20*x**2 + 25)/4
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {1}{4} \, \log \left (x^{10} + x^{9} + x^{8} - 2 \, x^{7} + 5 \, x^{6} - 4 \, x^{5} + 14 \, x^{4} - 10 \, x^{3} + 20 \, x^{2} + 25\right ) - 2 \, \log \left (x\right ) \] Input:
integrate((2*x^10+x^9+2*x^7-10*x^6+12*x^5-56*x^4+50*x^3-120*x^2-200)/(4*x^ 11+4*x^10+4*x^9-8*x^8+20*x^7-16*x^6+56*x^5-40*x^4+80*x^3+100*x),x, algorit hm="maxima")
Output:
1/4*log(x^10 + x^9 + x^8 - 2*x^7 + 5*x^6 - 4*x^5 + 14*x^4 - 10*x^3 + 20*x^ 2 + 25) - 2*log(x)
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {1}{4} \, \log \left (x^{10} + x^{9} + x^{8} - 2 \, x^{7} + 5 \, x^{6} - 4 \, x^{5} + 14 \, x^{4} - 10 \, x^{3} + 20 \, x^{2} + 25\right ) - 2 \, \log \left ({\left | x \right |}\right ) \] Input:
integrate((2*x^10+x^9+2*x^7-10*x^6+12*x^5-56*x^4+50*x^3-120*x^2-200)/(4*x^ 11+4*x^10+4*x^9-8*x^8+20*x^7-16*x^6+56*x^5-40*x^4+80*x^3+100*x),x, algorit hm="giac")
Output:
1/4*log(x^10 + x^9 + x^8 - 2*x^7 + 5*x^6 - 4*x^5 + 14*x^4 - 10*x^3 + 20*x^ 2 + 25) - 2*log(abs(x))
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {\ln \left (x^{10}+x^9+x^8-2\,x^7+5\,x^6-4\,x^5+14\,x^4-10\,x^3+20\,x^2+25\right )}{4}-2\,\ln \left (x\right ) \] Input:
int((50*x^3 - 120*x^2 - 56*x^4 + 12*x^5 - 10*x^6 + 2*x^7 + x^9 + 2*x^10 - 200)/(100*x + 80*x^3 - 40*x^4 + 56*x^5 - 16*x^6 + 20*x^7 - 8*x^8 + 4*x^9 + 4*x^10 + 4*x^11),x)
Output:
log(20*x^2 - 10*x^3 + 14*x^4 - 4*x^5 + 5*x^6 - 2*x^7 + x^8 + x^9 + x^10 + 25)/4 - 2*log(x)
Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {\mathrm {log}\left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}-2 \,\mathrm {log}\left (x \right ) \] Input:
int((2*x^10+x^9+2*x^7-10*x^6+12*x^5-56*x^4+50*x^3-120*x^2-200)/(4*x^11+4*x ^10+4*x^9-8*x^8+20*x^7-16*x^6+56*x^5-40*x^4+80*x^3+100*x),x)
Output:
(log(x**10 + x**9 + x**8 - 2*x**7 + 5*x**6 - 4*x**5 + 14*x**4 - 10*x**3 + 20*x**2 + 25) - 8*log(x))/4