Integrand size = 142, antiderivative size = 29 \[ \int \frac {625+523 x-48 x^2-e^{3 x} x^2+x^3+e^{2 x} \left (1250 x^2-102 x^3+2 x^4\right )+e^x \left (-1-2 x-390625 x^2+63750 x^3-3851 x^4+102 x^5-x^6\right )}{e^{3 x} x^2+e^{2 x} \left (-1250 x^2+102 x^3-2 x^4\right )+e^x \left (390625 x^2-63750 x^3+3851 x^4-102 x^5+x^6\right )} \, dx=-x+\frac {e^{-x}}{x \left (e^x-(25-x)^2+x\right )} \]
Time = 5.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {625+523 x-48 x^2-e^{3 x} x^2+x^3+e^{2 x} \left (1250 x^2-102 x^3+2 x^4\right )+e^x \left (-1-2 x-390625 x^2+63750 x^3-3851 x^4+102 x^5-x^6\right )}{e^{3 x} x^2+e^{2 x} \left (-1250 x^2+102 x^3-2 x^4\right )+e^x \left (390625 x^2-63750 x^3+3851 x^4-102 x^5+x^6\right )} \, dx=-\frac {e^{-x}-e^x x^2+x^2 \left (625-51 x+x^2\right )}{x \left (625-e^x-51 x+x^2\right )} \]
Integrate[(625 + 523*x - 48*x^2 - E^(3*x)*x^2 + x^3 + E^(2*x)*(1250*x^2 - 102*x^3 + 2*x^4) + E^x*(-1 - 2*x - 390625*x^2 + 63750*x^3 - 3851*x^4 + 102 *x^5 - x^6))/(E^(3*x)*x^2 + E^(2*x)*(-1250*x^2 + 102*x^3 - 2*x^4) + E^x*(3 90625*x^2 - 63750*x^3 + 3851*x^4 - 102*x^5 + x^6)),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.
\(\displaystyle \int \frac {x^3-e^{3 x} x^2-48 x^2+e^{2 x} \left (2 x^4-102 x^3+1250 x^2\right )+e^x \left (-x^6+102 x^5-3851 x^4+63750 x^3-390625 x^2-2 x-1\right )+523 x+625}{e^{3 x} x^2+e^{2 x} \left (-2 x^4+102 x^3-1250 x^2\right )+e^x \left (x^6-102 x^5+3851 x^4-63750 x^3+390625 x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-x} \left (x^3-e^{3 x} x^2-48 x^2+e^{2 x} \left (2 x^4-102 x^3+1250 x^2\right )+e^x \left (-x^6+102 x^5-3851 x^4+63750 x^3-390625 x^2-2 x-1\right )+523 x+625\right )}{x^2 \left (x^2-51 x-e^x+625\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{-x} (2 x+1)}{x^2 \left (x^2-51 x-e^x+625\right )}-\frac {e^{-x} \left (x^2-53 x+676\right )}{x \left (x^2-51 x-e^x+625\right )^2}-1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 53 \int \frac {e^{-x}}{\left (-x^2+51 x+e^x-625\right )^2}dx-676 \int \frac {e^{-x}}{x \left (x^2-51 x-e^x+625\right )^2}dx-\int \frac {e^{-x} x}{\left (x^2-51 x-e^x+625\right )^2}dx+\int \frac {e^{-x}}{x^2 \left (x^2-51 x-e^x+625\right )}dx+2 \int \frac {e^{-x}}{x \left (x^2-51 x-e^x+625\right )}dx-x\) |
Int[(625 + 523*x - 48*x^2 - E^(3*x)*x^2 + x^3 + E^(2*x)*(1250*x^2 - 102*x^ 3 + 2*x^4) + E^x*(-1 - 2*x - 390625*x^2 + 63750*x^3 - 3851*x^4 + 102*x^5 - x^6))/(E^(3*x)*x^2 + E^(2*x)*(-1250*x^2 + 102*x^3 - 2*x^4) + E^x*(390625* x^2 - 63750*x^3 + 3851*x^4 - 102*x^5 + x^6)),x]
3.17.58.3.1 Defintions of rubi rules used
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83
method | result | size |
risch | \(-x -\frac {{\mathrm e}^{-x}}{x \left (x^{2}-51 x +625\right )}-\frac {1}{x \left (x^{2}-51 x +625\right ) \left (x^{2}-51 x -{\mathrm e}^{x}+625\right )}\) | \(53\) |
parallelrisch | \(\frac {\left (-1-{\mathrm e}^{x} x^{4}+51 \,{\mathrm e}^{x} x^{3}+{\mathrm e}^{2 x} x^{2}-625 \,{\mathrm e}^{x} x^{2}\right ) {\mathrm e}^{-x}}{x \left (x^{2}-51 x -{\mathrm e}^{x}+625\right )}\) | \(54\) |
int((-x^2*exp(x)^3+(2*x^4-102*x^3+1250*x^2)*exp(x)^2+(-x^6+102*x^5-3851*x^ 4+63750*x^3-390625*x^2-2*x-1)*exp(x)+x^3-48*x^2+523*x+625)/(x^2*exp(x)^3+( -2*x^4+102*x^3-1250*x^2)*exp(x)^2+(x^6-102*x^5+3851*x^4-63750*x^3+390625*x ^2)*exp(x)),x,method=_RETURNVERBOSE)
Time = 0.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90 \[ \int \frac {625+523 x-48 x^2-e^{3 x} x^2+x^3+e^{2 x} \left (1250 x^2-102 x^3+2 x^4\right )+e^x \left (-1-2 x-390625 x^2+63750 x^3-3851 x^4+102 x^5-x^6\right )}{e^{3 x} x^2+e^{2 x} \left (-1250 x^2+102 x^3-2 x^4\right )+e^x \left (390625 x^2-63750 x^3+3851 x^4-102 x^5+x^6\right )} \, dx=-\frac {x^{2} e^{\left (2 \, x\right )} - {\left (x^{4} - 51 \, x^{3} + 625 \, x^{2}\right )} e^{x} - 1}{x e^{\left (2 \, x\right )} - {\left (x^{3} - 51 \, x^{2} + 625 \, x\right )} e^{x}} \]
integrate((-x^2*exp(x)^3+(2*x^4-102*x^3+1250*x^2)*exp(x)^2+(-x^6+102*x^5-3 851*x^4+63750*x^3-390625*x^2-2*x-1)*exp(x)+x^3-48*x^2+523*x+625)/(x^2*exp( x)^3+(-2*x^4+102*x^3-1250*x^2)*exp(x)^2+(x^6-102*x^5+3851*x^4-63750*x^3+39 0625*x^2)*exp(x)),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (17) = 34\).
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {625+523 x-48 x^2-e^{3 x} x^2+x^3+e^{2 x} \left (1250 x^2-102 x^3+2 x^4\right )+e^x \left (-1-2 x-390625 x^2+63750 x^3-3851 x^4+102 x^5-x^6\right )}{e^{3 x} x^2+e^{2 x} \left (-1250 x^2+102 x^3-2 x^4\right )+e^x \left (390625 x^2-63750 x^3+3851 x^4-102 x^5+x^6\right )} \, dx=- x + \frac {1}{- x^{5} + 102 x^{4} - 3851 x^{3} + 63750 x^{2} - 390625 x + \left (x^{3} - 51 x^{2} + 625 x\right ) e^{x}} - \frac {e^{- x}}{x^{3} - 51 x^{2} + 625 x} \]
integrate((-x**2*exp(x)**3+(2*x**4-102*x**3+1250*x**2)*exp(x)**2+(-x**6+10 2*x**5-3851*x**4+63750*x**3-390625*x**2-2*x-1)*exp(x)+x**3-48*x**2+523*x+6 25)/(x**2*exp(x)**3+(-2*x**4+102*x**3-1250*x**2)*exp(x)**2+(x**6-102*x**5+ 3851*x**4-63750*x**3+390625*x**2)*exp(x)),x)
-x + 1/(-x**5 + 102*x**4 - 3851*x**3 + 63750*x**2 - 390625*x + (x**3 - 51* x**2 + 625*x)*exp(x)) - exp(-x)/(x**3 - 51*x**2 + 625*x)
Time = 0.27 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.10 \[ \int \frac {625+523 x-48 x^2-e^{3 x} x^2+x^3+e^{2 x} \left (1250 x^2-102 x^3+2 x^4\right )+e^x \left (-1-2 x-390625 x^2+63750 x^3-3851 x^4+102 x^5-x^6\right )}{e^{3 x} x^2+e^{2 x} \left (-1250 x^2+102 x^3-2 x^4\right )+e^x \left (390625 x^2-63750 x^3+3851 x^4-102 x^5+x^6\right )} \, dx=-x \]
integrate((-x^2*exp(x)^3+(2*x^4-102*x^3+1250*x^2)*exp(x)^2+(-x^6+102*x^5-3 851*x^4+63750*x^3-390625*x^2-2*x-1)*exp(x)+x^3-48*x^2+523*x+625)/(x^2*exp( x)^3+(-2*x^4+102*x^3-1250*x^2)*exp(x)^2+(x^6-102*x^5+3851*x^4-63750*x^3+39 0625*x^2)*exp(x)),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10 \[ \int \frac {625+523 x-48 x^2-e^{3 x} x^2+x^3+e^{2 x} \left (1250 x^2-102 x^3+2 x^4\right )+e^x \left (-1-2 x-390625 x^2+63750 x^3-3851 x^4+102 x^5-x^6\right )}{e^{3 x} x^2+e^{2 x} \left (-1250 x^2+102 x^3-2 x^4\right )+e^x \left (390625 x^2-63750 x^3+3851 x^4-102 x^5+x^6\right )} \, dx=-\frac {x^{4} e^{x} - 51 \, x^{3} e^{x} - x^{2} e^{\left (2 \, x\right )} + 625 \, x^{2} e^{x} + 1}{x^{3} e^{x} - 51 \, x^{2} e^{x} - x e^{\left (2 \, x\right )} + 625 \, x e^{x}} \]
integrate((-x^2*exp(x)^3+(2*x^4-102*x^3+1250*x^2)*exp(x)^2+(-x^6+102*x^5-3 851*x^4+63750*x^3-390625*x^2-2*x-1)*exp(x)+x^3-48*x^2+523*x+625)/(x^2*exp( x)^3+(-2*x^4+102*x^3-1250*x^2)*exp(x)^2+(x^6-102*x^5+3851*x^4-63750*x^3+39 0625*x^2)*exp(x)),x, algorithm=\
-(x^4*e^x - 51*x^3*e^x - x^2*e^(2*x) + 625*x^2*e^x + 1)/(x^3*e^x - 51*x^2* e^x - x*e^(2*x) + 625*x*e^x)
Time = 14.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {625+523 x-48 x^2-e^{3 x} x^2+x^3+e^{2 x} \left (1250 x^2-102 x^3+2 x^4\right )+e^x \left (-1-2 x-390625 x^2+63750 x^3-3851 x^4+102 x^5-x^6\right )}{e^{3 x} x^2+e^{2 x} \left (-1250 x^2+102 x^3-2 x^4\right )+e^x \left (390625 x^2-63750 x^3+3851 x^4-102 x^5+x^6\right )} \, dx=\frac {{\mathrm {e}}^{-x}\,\left (625\,x^2\,{\mathrm {e}}^x-51\,x^3\,{\mathrm {e}}^x+x^4\,{\mathrm {e}}^x-x^2\,{\mathrm {e}}^{2\,x}+1\right )}{x\,\left (51\,x+{\mathrm {e}}^x-x^2-625\right )} \]
int((523*x - exp(x)*(2*x + 390625*x^2 - 63750*x^3 + 3851*x^4 - 102*x^5 + x ^6 + 1) - x^2*exp(3*x) + exp(2*x)*(1250*x^2 - 102*x^3 + 2*x^4) - 48*x^2 + x^3 + 625)/(exp(x)*(390625*x^2 - 63750*x^3 + 3851*x^4 - 102*x^5 + x^6) + x ^2*exp(3*x) - exp(2*x)*(1250*x^2 - 102*x^3 + 2*x^4)),x)