Integrand size = 65, antiderivative size = 22 \[ \int \frac {2375+4 e^4+e^2 (-195-8 x)+200 x+4 x^2}{2500+2575 x+199 x^2+4 x^3+e^4 (4+4 x)+e^2 \left (-200-203 x-8 x^2\right )} \, dx=5+\log \left (-4+x-5 \left (x+\frac {x}{-25+e^2-x}\right )\right ) \] Output:
5+ln(-4*x-4-5*x/(exp(2)-x-25))
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {2375+4 e^4+e^2 (-195-8 x)+200 x+4 x^2}{2500+2575 x+199 x^2+4 x^3+e^4 (4+4 x)+e^2 \left (-200-203 x-8 x^2\right )} \, dx=-\log \left (25-e^2+x\right )+\log \left (100-4 e^2+99 x-4 e^2 x+4 x^2\right ) \] Input:
Integrate[(2375 + 4*E^4 + E^2*(-195 - 8*x) + 200*x + 4*x^2)/(2500 + 2575*x + 199*x^2 + 4*x^3 + E^4*(4 + 4*x) + E^2*(-200 - 203*x - 8*x^2)),x]
Output:
-Log[25 - E^2 + x] + Log[100 - 4*E^2 + 99*x - 4*E^2*x + 4*x^2]
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2462, 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.
\(\displaystyle \int \frac {4 x^2+200 x+e^2 (-8 x-195)+4 e^4+2375}{4 x^3+199 x^2+e^2 \left (-8 x^2-203 x-200\right )+2575 x+e^4 (4 x+4)+2500} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {8 x-4 e^2+99}{4 x^2+\left (99-4 e^2\right ) x+4 \left (25-e^2\right )}+\frac {1}{-x+e^2-25}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (4 x^2+\left (99-4 e^2\right ) x+4 \left (25-e^2\right )\right )-\log \left (x-e^2+25\right )\) |
Input:
Int[(2375 + 4*E^4 + E^2*(-195 - 8*x) + 200*x + 4*x^2)/(2500 + 2575*x + 199 *x^2 + 4*x^3 + E^4*(4 + 4*x) + E^2*(-200 - 203*x - 8*x^2)),x]
Output:
-Log[25 - E^2 + x] + Log[4*(25 - E^2) + (99 - 4*E^2)*x + 4*x^2]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36
method | result | size |
parallelrisch | \(-\ln \left (-{\mathrm e}^{2}+x +25\right )+\ln \left (-{\mathrm e}^{2} x +x^{2}-{\mathrm e}^{2}+\frac {99 x}{4}+25\right )\) | \(30\) |
norman | \(-\ln \left ({\mathrm e}^{2}-x -25\right )+\ln \left (4 \,{\mathrm e}^{2} x -4 x^{2}+4 \,{\mathrm e}^{2}-99 x -100\right )\) | \(32\) |
risch | \(-\ln \left (-{\mathrm e}^{2}+x +25\right )+\ln \left (4 x^{2}+\left (-4 \,{\mathrm e}^{2}+99\right ) x -4 \,{\mathrm e}^{2}+100\right )\) | \(32\) |
default | \(-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{3}+\left (-8 \,{\mathrm e}^{2}+199\right ) \textit {\_Z}^{2}+\left (-203 \,{\mathrm e}^{2}+4 \,{\mathrm e}^{4}+2575\right ) \textit {\_Z} -200 \,{\mathrm e}^{2}+4 \,{\mathrm e}^{4}+2500\right )}{\sum }\frac {\left (-4 \,{\mathrm e}^{4}+8 \,{\mathrm e}^{2} \textit {\_R} -4 \textit {\_R}^{2}+195 \,{\mathrm e}^{2}-200 \textit {\_R} -2375\right ) \ln \left (x -\textit {\_R} \right )}{4 \,{\mathrm e}^{4}-16 \,{\mathrm e}^{2} \textit {\_R} +12 \textit {\_R}^{2}-203 \,{\mathrm e}^{2}+398 \textit {\_R} +2575}\right )\) | \(99\) |
Input:
int((4*exp(2)^2+(-8*x-195)*exp(2)+4*x^2+200*x+2375)/((4+4*x)*exp(2)^2+(-8* x^2-203*x-200)*exp(2)+4*x^3+199*x^2+2575*x+2500),x,method=_RETURNVERBOSE)
Output:
-ln(-exp(2)+x+25)+ln(-exp(2)*x+x^2-exp(2)+99/4*x+25)
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {2375+4 e^4+e^2 (-195-8 x)+200 x+4 x^2}{2500+2575 x+199 x^2+4 x^3+e^4 (4+4 x)+e^2 \left (-200-203 x-8 x^2\right )} \, dx=\log \left (4 \, x^{2} - 4 \, {\left (x + 1\right )} e^{2} + 99 \, x + 100\right ) - \log \left (x - e^{2} + 25\right ) \] Input:
integrate((4*exp(2)^2+(-8*x-195)*exp(2)+4*x^2+200*x+2375)/((4+4*x)*exp(2)^ 2+(-8*x^2-203*x-200)*exp(2)+4*x^3+199*x^2+2575*x+2500),x, algorithm="frica s")
Output:
log(4*x^2 - 4*(x + 1)*e^2 + 99*x + 100) - log(x - e^2 + 25)
Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {2375+4 e^4+e^2 (-195-8 x)+200 x+4 x^2}{2500+2575 x+199 x^2+4 x^3+e^4 (4+4 x)+e^2 \left (-200-203 x-8 x^2\right )} \, dx=- \log {\left (x - e^{2} + 25 \right )} + \log {\left (x^{2} + x \left (\frac {99}{4} - e^{2}\right ) - e^{2} + 25 \right )} \] Input:
integrate((4*exp(2)**2+(-8*x-195)*exp(2)+4*x**2+200*x+2375)/((4+4*x)*exp(2 )**2+(-8*x**2-203*x-200)*exp(2)+4*x**3+199*x**2+2575*x+2500),x)
Output:
-log(x - exp(2) + 25) + log(x**2 + x*(99/4 - exp(2)) - exp(2) + 25)
Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {2375+4 e^4+e^2 (-195-8 x)+200 x+4 x^2}{2500+2575 x+199 x^2+4 x^3+e^4 (4+4 x)+e^2 \left (-200-203 x-8 x^2\right )} \, dx=\log \left (4 \, x^{2} - x {\left (4 \, e^{2} - 99\right )} - 4 \, e^{2} + 100\right ) - \log \left (x - e^{2} + 25\right ) \] Input:
integrate((4*exp(2)^2+(-8*x-195)*exp(2)+4*x^2+200*x+2375)/((4+4*x)*exp(2)^ 2+(-8*x^2-203*x-200)*exp(2)+4*x^3+199*x^2+2575*x+2500),x, algorithm="maxim a")
Output:
log(4*x^2 - x*(4*e^2 - 99) - 4*e^2 + 100) - log(x - e^2 + 25)
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {2375+4 e^4+e^2 (-195-8 x)+200 x+4 x^2}{2500+2575 x+199 x^2+4 x^3+e^4 (4+4 x)+e^2 \left (-200-203 x-8 x^2\right )} \, dx=\log \left ({\left | 4 \, x^{2} - 4 \, x e^{2} + 99 \, x - 4 \, e^{2} + 100 \right |}\right ) - \log \left ({\left | x - e^{2} + 25 \right |}\right ) \] Input:
integrate((4*exp(2)^2+(-8*x-195)*exp(2)+4*x^2+200*x+2375)/((4+4*x)*exp(2)^ 2+(-8*x^2-203*x-200)*exp(2)+4*x^3+199*x^2+2575*x+2500),x, algorithm="giac" )
Output:
log(abs(4*x^2 - 4*x*e^2 + 99*x - 4*e^2 + 100)) - log(abs(x - e^2 + 25))
Time = 3.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {2375+4 e^4+e^2 (-195-8 x)+200 x+4 x^2}{2500+2575 x+199 x^2+4 x^3+e^4 (4+4 x)+e^2 \left (-200-203 x-8 x^2\right )} \, dx=\ln \left (\frac {99\,x}{4}-{\mathrm {e}}^2-x\,{\mathrm {e}}^2+x^2+25\right )-\ln \left (x-{\mathrm {e}}^2+25\right ) \] Input:
int((200*x + 4*exp(4) + 4*x^2 - exp(2)*(8*x + 195) + 2375)/(2575*x - exp(2 )*(203*x + 8*x^2 + 200) + 199*x^2 + 4*x^3 + exp(4)*(4*x + 4) + 2500),x)
Output:
log((99*x)/4 - exp(2) - x*exp(2) + x^2 + 25) - log(x - exp(2) + 25)
\[ \int \frac {2375+4 e^4+e^2 (-195-8 x)+200 x+4 x^2}{2500+2575 x+199 x^2+4 x^3+e^4 (4+4 x)+e^2 \left (-200-203 x-8 x^2\right )} \, dx=-\frac {8 \left (\int \frac {x}{4 e^{4} x +4 e^{4}-8 e^{2} x^{2}-203 e^{2} x +4 x^{3}-200 e^{2}+199 x^{2}+2575 x +2500}d x \right ) e^{2}}{3}+\frac {202 \left (\int \frac {x}{4 e^{4} x +4 e^{4}-8 e^{2} x^{2}-203 e^{2} x +4 x^{3}-200 e^{2}+199 x^{2}+2575 x +2500}d x \right )}{3}+\frac {8 \left (\int \frac {1}{4 e^{4} x +4 e^{4}-8 e^{2} x^{2}-203 e^{2} x +4 x^{3}-200 e^{2}+199 x^{2}+2575 x +2500}d x \right ) e^{4}}{3}-\frac {382 \left (\int \frac {1}{4 e^{4} x +4 e^{4}-8 e^{2} x^{2}-203 e^{2} x +4 x^{3}-200 e^{2}+199 x^{2}+2575 x +2500}d x \right ) e^{2}}{3}+\frac {4550 \left (\int \frac {1}{4 e^{4} x +4 e^{4}-8 e^{2} x^{2}-203 e^{2} x +4 x^{3}-200 e^{2}+199 x^{2}+2575 x +2500}d x \right )}{3}+\frac {\mathrm {log}\left (4 e^{4} x +4 e^{4}-8 e^{2} x^{2}-203 e^{2} x +4 x^{3}-200 e^{2}+199 x^{2}+2575 x +2500\right )}{3} \] Input:
int((4*exp(2)^2+(-8*x-195)*exp(2)+4*x^2+200*x+2375)/((4+4*x)*exp(2)^2+(-8* x^2-203*x-200)*exp(2)+4*x^3+199*x^2+2575*x+2500),x)
Output:
( - 8*int(x/(4*e**4*x + 4*e**4 - 8*e**2*x**2 - 203*e**2*x - 200*e**2 + 4*x **3 + 199*x**2 + 2575*x + 2500),x)*e**2 + 202*int(x/(4*e**4*x + 4*e**4 - 8 *e**2*x**2 - 203*e**2*x - 200*e**2 + 4*x**3 + 199*x**2 + 2575*x + 2500),x) + 8*int(1/(4*e**4*x + 4*e**4 - 8*e**2*x**2 - 203*e**2*x - 200*e**2 + 4*x* *3 + 199*x**2 + 2575*x + 2500),x)*e**4 - 382*int(1/(4*e**4*x + 4*e**4 - 8* e**2*x**2 - 203*e**2*x - 200*e**2 + 4*x**3 + 199*x**2 + 2575*x + 2500),x)* e**2 + 4550*int(1/(4*e**4*x + 4*e**4 - 8*e**2*x**2 - 203*e**2*x - 200*e**2 + 4*x**3 + 199*x**2 + 2575*x + 2500),x) + log(4*e**4*x + 4*e**4 - 8*e**2* x**2 - 203*e**2*x - 200*e**2 + 4*x**3 + 199*x**2 + 2575*x + 2500))/3