Integrand size = 99, antiderivative size = 32 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=3-(1-x) \left (3 x+\frac {x}{-e^x+\frac {5}{x+x^2}}\right ) \] Output:
3-(1-x)*(x/(5/(x^2+x)-exp(x))+3*x)
Time = 4.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=x \left (-3+3 x+\frac {x-x^3}{-5+e^x x (1+x)}\right ) \] Input:
Integrate[(-75 + 140*x + 20*x^3 + E^(2*x)*(-3*x^2 + 9*x^4 + 6*x^5) + E^x*( 30*x - 29*x^2 - 61*x^3 - 4*x^4 - x^5 + x^6))/(25 + E^x*(-10*x - 10*x^2) + E^(2*x)*(x^2 + 2*x^3 + x^4)),x]
Output:
x*(-3 + 3*x + (x - x^3)/(-5 + E^x*x*(1 + 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.
\(\displaystyle \int \frac {20 x^3+e^{2 x} \left (6 x^5+9 x^4-3 x^2\right )+e^x \left (x^6-x^5-4 x^4-61 x^3-29 x^2+30 x\right )+140 x-75}{e^x \left (-10 x^2-10 x\right )+e^{2 x} \left (x^4+2 x^3+x^2\right )+25} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {20 x^3+e^{2 x} \left (6 x^5+9 x^4-3 x^2\right )+e^x \left (x^6-x^5-4 x^4-61 x^3-29 x^2+30 x\right )+140 x-75}{\left (-e^x x^2-e^x x+5\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x \left (x^3-2 x^2-2 x+1\right )}{e^x x^2+e^x x-5}+\frac {5 x \left (x^3+2 x^2-2 x-1\right )}{\left (e^x x^2+e^x x-5\right )^2}+3 (2 x-1)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -5 \int \frac {x}{\left (e^x x^2+e^x x-5\right )^2}dx-10 \int \frac {x^2}{\left (e^x x^2+e^x x-5\right )^2}dx+\int \frac {x}{e^x x^2+e^x x-5}dx-2 \int \frac {x^2}{e^x x^2+e^x x-5}dx+5 \int \frac {x^4}{\left (e^x x^2+e^x x-5\right )^2}dx+\int \frac {x^4}{e^x x^2+e^x x-5}dx+10 \int \frac {x^3}{\left (e^x x^2+e^x x-5\right )^2}dx-2 \int \frac {x^3}{e^x x^2+e^x x-5}dx+\frac {3}{4} (1-2 x)^2\) |
Input:
Int[(-75 + 140*x + 20*x^3 + E^(2*x)*(-3*x^2 + 9*x^4 + 6*x^5) + E^x*(30*x - 29*x^2 - 61*x^3 - 4*x^4 - x^5 + x^6))/(25 + E^x*(-10*x - 10*x^2) + E^(2*x )*(x^2 + 2*x^3 + x^4)),x]
Output:
$Aborted
Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09
method | result | size |
risch | \(3 x^{2}-3 x -\frac {\left (-1+x \right ) x^{2} \left (1+x \right )}{{\mathrm e}^{x} x^{2}+{\mathrm e}^{x} x -5}\) | \(35\) |
norman | \(\frac {3 \,{\mathrm e}^{x} x +15 x -14 x^{2}-x^{4}+3 \,{\mathrm e}^{x} x^{4}-15}{{\mathrm e}^{x} x^{2}+{\mathrm e}^{x} x -5}\) | \(43\) |
parallelrisch | \(\frac {3 \,{\mathrm e}^{x} x +15 x -14 x^{2}-x^{4}+3 \,{\mathrm e}^{x} x^{4}-15}{{\mathrm e}^{x} x^{2}+{\mathrm e}^{x} x -5}\) | \(43\) |
Input:
int(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x)*exp(x )+20*x^3+140*x-75)/((x^4+2*x^3+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+25),x,m ethod=_RETURNVERBOSE)
Output:
3*x^2-3*x-(-1+x)*x^2*(1+x)/(exp(x)*x^2+exp(x)*x-5)
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=-\frac {x^{4} + 14 \, x^{2} - 3 \, {\left (x^{4} - x^{2}\right )} e^{x} - 15 \, x}{{\left (x^{2} + x\right )} e^{x} - 5} \] Input:
integrate(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x) *exp(x)+20*x^3+140*x-75)/((x^4+2*x^3+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+2 5),x, algorithm="fricas")
Output:
-(x^4 + 14*x^2 - 3*(x^4 - x^2)*e^x - 15*x)/((x^2 + x)*e^x - 5)
Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=3 x^{2} - 3 x + \frac {- x^{4} + x^{2}}{\left (x^{2} + x\right ) e^{x} - 5} \] Input:
integrate(((6*x**5+9*x**4-3*x**2)*exp(x)**2+(x**6-x**5-4*x**4-61*x**3-29*x **2+30*x)*exp(x)+20*x**3+140*x-75)/((x**4+2*x**3+x**2)*exp(x)**2+(-10*x**2 -10*x)*exp(x)+25),x)
Output:
3*x**2 - 3*x + (-x**4 + x**2)/((x**2 + x)*exp(x) - 5)
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=-\frac {x^{4} + 14 \, x^{2} - 3 \, {\left (x^{4} - x^{2}\right )} e^{x} - 15 \, x}{{\left (x^{2} + x\right )} e^{x} - 5} \] Input:
integrate(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x) *exp(x)+20*x^3+140*x-75)/((x^4+2*x^3+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+2 5),x, algorithm="maxima")
Output:
-(x^4 + 14*x^2 - 3*(x^4 - x^2)*e^x - 15*x)/((x^2 + x)*e^x - 5)
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=\frac {3 \, x^{4} e^{x} - x^{4} - 3 \, x^{2} e^{x} - 14 \, x^{2} + 15 \, x}{x^{2} e^{x} + x e^{x} - 5} \] Input:
integrate(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x) *exp(x)+20*x^3+140*x-75)/((x^4+2*x^3+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+2 5),x, algorithm="giac")
Output:
(3*x^4*e^x - x^4 - 3*x^2*e^x - 14*x^2 + 15*x)/(x^2*e^x + x*e^x - 5)
Time = 3.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=-\frac {x\,\left (x-1\right )\,\left (x-3\,x^2\,{\mathrm {e}}^x-3\,x\,{\mathrm {e}}^x+x^2+15\right )}{x^2\,{\mathrm {e}}^x+x\,{\mathrm {e}}^x-5} \] Input:
int((140*x - exp(x)*(29*x^2 - 30*x + 61*x^3 + 4*x^4 + x^5 - x^6) + exp(2*x )*(9*x^4 - 3*x^2 + 6*x^5) + 20*x^3 - 75)/(exp(2*x)*(x^2 + 2*x^3 + x^4) - e xp(x)*(10*x + 10*x^2) + 25),x)
Output:
-(x*(x - 1)*(x - 3*x^2*exp(x) - 3*x*exp(x) + x^2 + 15))/(x^2*exp(x) + x*ex p(x) - 5)
Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=\frac {x \left (3 e^{x} x^{3}-3 e^{x} x -x^{3}-14 x +15\right )}{e^{x} x^{2}+e^{x} x -5} \] Input:
int(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x)*exp(x )+20*x^3+140*x-75)/((x^4+2*x^3+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+25),x)
Output:
(x*(3*e**x*x**3 - 3*e**x*x - x**3 - 14*x + 15))/(e**x*x**2 + e**x*x - 5)