Integrand size = 87, antiderivative size = 34 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=-1+x+\frac {e^x x}{\left (-x+25 x^2\right ) \left (-1+\frac {x^2}{2-x}\right )} \] Output:
x+x/(25*x^2-x)/(x^2/(2-x)-1)*exp(x)-1
Time = 0.82 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=x+\frac {e^x (2-x)}{2-51 x+24 x^2+25 x^3} \] Input:
Integrate[(4 - 204*x + 2697*x^2 - 2348*x^3 - 1974*x^4 + 1200*x^5 + 625*x^6 + E^x*(104 - 200*x - 27*x^2 + 76*x^3 - 25*x^4))/(4 - 204*x + 2697*x^2 - 2 348*x^3 - 1974*x^4 + 1200*x^5 + 625*x^6),x]
Output:
x + (E^x*(2 - x))/(2 - 51*x + 24*x^2 + 25*x^3)
Time = 1.96 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2463, 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.
\(\displaystyle \int \frac {625 x^6+1200 x^5-1974 x^4-2348 x^3+2697 x^2+e^x \left (-25 x^4+76 x^3-27 x^2-200 x+104\right )-204 x+4}{625 x^6+1200 x^5-1974 x^4-2348 x^3+2697 x^2-204 x+4} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {11 \left (625 x^6+1200 x^5-1974 x^4-2348 x^3+2697 x^2+e^x \left (-25 x^4+76 x^3-27 x^2-200 x+104\right )-204 x+4\right )}{20736 (x-1)}+\frac {28 \left (625 x^6+1200 x^5-1974 x^4-2348 x^3+2697 x^2+e^x \left (-25 x^4+76 x^3-27 x^2-200 x+104\right )-204 x+4\right )}{397953 (x+2)}+\frac {390625 \left (625 x^6+1200 x^5-1974 x^4-2348 x^3+2697 x^2+e^x \left (-25 x^4+76 x^3-27 x^2-200 x+104\right )-204 x+4\right )}{33958656 (25 x-1)}+\frac {625 x^6+1200 x^5-1974 x^4-2348 x^3+2697 x^2+e^x \left (-25 x^4+76 x^3-27 x^2-200 x+104\right )-204 x+4}{5184 (x-1)^2}+\frac {625 x^6+1200 x^5-1974 x^4-2348 x^3+2697 x^2+e^x \left (-25 x^4+76 x^3-27 x^2-200 x+104\right )-204 x+4}{23409 (x+2)^2}+\frac {390625 \left (625 x^6+1200 x^5-1974 x^4-2348 x^3+2697 x^2+e^x \left (-25 x^4+76 x^3-27 x^2-200 x+104\right )-204 x+4\right )}{1498176 (25 x-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x+\frac {1225 e^x}{1224 (1-25 x)}-\frac {e^x}{72 (1-x)}+\frac {4 e^x}{153 (x+2)}\) |
Input:
Int[(4 - 204*x + 2697*x^2 - 2348*x^3 - 1974*x^4 + 1200*x^5 + 625*x^6 + E^x *(104 - 200*x - 27*x^2 + 76*x^3 - 25*x^4))/(4 - 204*x + 2697*x^2 - 2348*x^ 3 - 1974*x^4 + 1200*x^5 + 625*x^6),x]
Output:
(1225*E^x)/(1224*(1 - 25*x)) - E^x/(72*(1 - x)) + x + (4*E^x)/(153*(2 + x) )
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] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 1.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79
method | result | size |
risch | \(x -\frac {\left (-2+x \right ) {\mathrm e}^{x}}{25 x^{3}+24 x^{2}-51 x +2}\) | \(27\) |
default | \(x +\frac {4 \,{\mathrm e}^{x}}{153 \left (2+x \right )}-\frac {49 \,{\mathrm e}^{x}}{1224 \left (x -\frac {1}{25}\right )}+\frac {{\mathrm e}^{x}}{-72+72 x}\) | \(30\) |
parts | \(x +\frac {4 \,{\mathrm e}^{x}}{153 \left (2+x \right )}-\frac {49 \,{\mathrm e}^{x}}{1224 \left (x -\frac {1}{25}\right )}+\frac {{\mathrm e}^{x}}{-72+72 x}\) | \(30\) |
norman | \(\frac {-\frac {1851 x^{2}}{25}+\frac {1274 x}{25}+25 x^{4}-{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}-\frac {48}{25}}{25 x^{3}+24 x^{2}-51 x +2}\) | \(43\) |
parallelrisch | \(\frac {625 x^{4}-48-1851 x^{2}-25 \,{\mathrm e}^{x} x +1274 x +50 \,{\mathrm e}^{x}}{625 x^{3}+600 x^{2}-1275 x +50}\) | \(44\) |
orering | \(\frac {\left (x +\frac {101}{25}\right ) \left (\left (-25 x^{4}+76 x^{3}-27 x^{2}-200 x +104\right ) {\mathrm e}^{x}+625 x^{6}+1200 x^{5}-1974 x^{4}-2348 x^{3}+2697 x^{2}-204 x +4\right )}{625 x^{6}+1200 x^{5}-1974 x^{4}-2348 x^{3}+2697 x^{2}-204 x +4}-\frac {\left (-1+x \right ) \left (2+x \right ) \left (25 x -1\right ) \left (625 x^{5}-6351 x^{3}+10202 x^{2}+15000 x -10404\right ) \left (\frac {\left (-100 x^{3}+228 x^{2}-54 x -200\right ) {\mathrm e}^{x}+\left (-25 x^{4}+76 x^{3}-27 x^{2}-200 x +104\right ) {\mathrm e}^{x}+3750 x^{5}+6000 x^{4}-7896 x^{3}-7044 x^{2}+5394 x -204}{625 x^{6}+1200 x^{5}-1974 x^{4}-2348 x^{3}+2697 x^{2}-204 x +4}-\frac {\left (\left (-25 x^{4}+76 x^{3}-27 x^{2}-200 x +104\right ) {\mathrm e}^{x}+625 x^{6}+1200 x^{5}-1974 x^{4}-2348 x^{3}+2697 x^{2}-204 x +4\right ) \left (3750 x^{5}+6000 x^{4}-7896 x^{3}-7044 x^{2}+5394 x -204\right )}{\left (625 x^{6}+1200 x^{5}-1974 x^{4}-2348 x^{3}+2697 x^{2}-204 x +4\right )^{2}}\right )}{25 \left (625 x^{7}-2550 x^{6}+3276 x^{5}+6148 x^{4}-21549 x^{3}-11898 x^{2}+25996 x -10416\right )}\) | \(373\) |
Input:
int(((-25*x^4+76*x^3-27*x^2-200*x+104)*exp(x)+625*x^6+1200*x^5-1974*x^4-23 48*x^3+2697*x^2-204*x+4)/(625*x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^2-204* x+4),x,method=_RETURNVERBOSE)
Output:
x-(-2+x)/(25*x^3+24*x^2-51*x+2)*exp(x)
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=\frac {25 \, x^{4} + 24 \, x^{3} - 51 \, x^{2} - {\left (x - 2\right )} e^{x} + 2 \, x}{25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2} \] Input:
integrate(((-25*x^4+76*x^3-27*x^2-200*x+104)*exp(x)+625*x^6+1200*x^5-1974* x^4-2348*x^3+2697*x^2-204*x+4)/(625*x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^ 2-204*x+4),x, algorithm="fricas")
Output:
(25*x^4 + 24*x^3 - 51*x^2 - (x - 2)*e^x + 2*x)/(25*x^3 + 24*x^2 - 51*x + 2 )
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=x + \frac {\left (2 - x\right ) e^{x}}{25 x^{3} + 24 x^{2} - 51 x + 2} \] Input:
integrate(((-25*x**4+76*x**3-27*x**2-200*x+104)*exp(x)+625*x**6+1200*x**5- 1974*x**4-2348*x**3+2697*x**2-204*x+4)/(625*x**6+1200*x**5-1974*x**4-2348* x**3+2697*x**2-204*x+4),x)
Output:
x + (2 - x)*exp(x)/(25*x**3 + 24*x**2 - 51*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (32) = 64\).
Time = 0.10 (sec) , antiderivative size = 229, normalized size of antiderivative = 6.74 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=x - \frac {{\left (x - 2\right )} e^{x}}{25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2} - \frac {285481771 \, x^{2} - 240455729 \, x + 9161458}{6242400 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} + \frac {4580729 \, x^{2} - 7021771 \, x + 273542}{130050 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} + \frac {329 \, {\left (136771 \, x^{2} - 51929 \, x + 1858\right )}}{1040400 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} - \frac {4075 \, x^{2} + 4687 \, x - 5294}{62424 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} - \frac {899 \, {\left (2275 \, x^{2} + 1255 \, x - 62\right )}}{83232 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} - \frac {587 \, {\left (929 \, x^{2} - 4579 \, x + 182\right )}}{62424 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} + \frac {775 \, x^{2} + 3019 \, x - 326}{1224 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} \] Input:
integrate(((-25*x^4+76*x^3-27*x^2-200*x+104)*exp(x)+625*x^6+1200*x^5-1974* x^4-2348*x^3+2697*x^2-204*x+4)/(625*x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^ 2-204*x+4),x, algorithm="maxima")
Output:
x - (x - 2)*e^x/(25*x^3 + 24*x^2 - 51*x + 2) - 1/6242400*(285481771*x^2 - 240455729*x + 9161458)/(25*x^3 + 24*x^2 - 51*x + 2) + 1/130050*(4580729*x^ 2 - 7021771*x + 273542)/(25*x^3 + 24*x^2 - 51*x + 2) + 329/1040400*(136771 *x^2 - 51929*x + 1858)/(25*x^3 + 24*x^2 - 51*x + 2) - 1/62424*(4075*x^2 + 4687*x - 5294)/(25*x^3 + 24*x^2 - 51*x + 2) - 899/83232*(2275*x^2 + 1255*x - 62)/(25*x^3 + 24*x^2 - 51*x + 2) - 587/62424*(929*x^2 - 4579*x + 182)/( 25*x^3 + 24*x^2 - 51*x + 2) + 1/1224*(775*x^2 + 3019*x - 326)/(25*x^3 + 24 *x^2 - 51*x + 2)
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=\frac {25 \, x^{4} + 24 \, x^{3} - 51 \, x^{2} - x e^{x} + 2 \, x + 2 \, e^{x}}{25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2} \] Input:
integrate(((-25*x^4+76*x^3-27*x^2-200*x+104)*exp(x)+625*x^6+1200*x^5-1974* x^4-2348*x^3+2697*x^2-204*x+4)/(625*x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^ 2-204*x+4),x, algorithm="giac")
Output:
(25*x^4 + 24*x^3 - 51*x^2 - x*e^x + 2*x + 2*e^x)/(25*x^3 + 24*x^2 - 51*x + 2)
Time = 7.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=x+\frac {2\,{\mathrm {e}}^x-x\,{\mathrm {e}}^x}{\left (25\,x-1\right )\,\left (x-1\right )\,\left (x+2\right )} \] Input:
int(-(204*x + exp(x)*(200*x + 27*x^2 - 76*x^3 + 25*x^4 - 104) - 2697*x^2 + 2348*x^3 + 1974*x^4 - 1200*x^5 - 625*x^6 - 4)/(2697*x^2 - 204*x - 2348*x^ 3 - 1974*x^4 + 1200*x^5 + 625*x^6 + 4),x)
Output:
x + (2*exp(x) - x*exp(x))/((25*x - 1)*(x - 1)*(x + 2))
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=\frac {-e^{x} x +2 e^{x}+25 x^{4}+24 x^{3}-51 x^{2}+2 x}{25 x^{3}+24 x^{2}-51 x +2} \] Input:
int(((-25*x^4+76*x^3-27*x^2-200*x+104)*exp(x)+625*x^6+1200*x^5-1974*x^4-23 48*x^3+2697*x^2-204*x+4)/(625*x^6+1200*x^5-1974*x^4-2348*x^3+2697*x^2-204* x+4),x)
Output:
( - e**x*x + 2*e**x + 25*x**4 + 24*x**3 - 51*x**2 + 2*x)/(25*x**3 + 24*x** 2 - 51*x + 2)