Integrand size = 130, antiderivative size = 22 \[ \int \frac {e^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (-x+625 x^2\right )+e^x \left (x^2+x^3\right )+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} \left (2 x+390625 x^2\right )+\left (2 e^{3 x}+2 e^x x+1250 e^{2 x} x\right ) \log (x)+e^{2 x} \log ^2(x)} \, dx=\frac {x^2}{e^x+625 x+e^{-x} x+\log (x)} \] Output:
x^2/(625*x+x/exp(x)+ln(x)+exp(x))
Time = 3.44 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {e^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (-x+625 x^2\right )+e^x \left (x^2+x^3\right )+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} \left (2 x+390625 x^2\right )+\left (2 e^{3 x}+2 e^x x+1250 e^{2 x} x\right ) \log (x)+e^{2 x} \log ^2(x)} \, dx=\frac {e^x x^2}{e^{2 x}+x+625 e^x x+e^x \log (x)} \] Input:
Integrate[(E^(3*x)*(2*x - x^2) + E^(2*x)*(-x + 625*x^2) + E^x*(x^2 + x^3) + 2*E^(2*x)*x*Log[x])/(E^(4*x) + 1250*E^(3*x)*x + x^2 + 1250*E^x*x^2 + E^( 2*x)*(2*x + 390625*x^2) + (2*E^(3*x) + 2*E^x*x + 1250*E^(2*x)*x)*Log[x] + E^(2*x)*Log[x]^2),x]
Output:
(E^x*x^2)/(E^(2*x) + x + 625*E^x*x + E^x*Log[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.
\(\displaystyle \int \frac {e^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (625 x^2-x\right )+e^x \left (x^3+x^2\right )+2 e^{2 x} x \log (x)}{1250 e^x x^2+x^2+e^{2 x} \left (390625 x^2+2 x\right )+1250 e^{3 x} x+e^{4 x}+e^{2 x} \log ^2(x)+\left (2 e^x x+1250 e^{2 x} x+2 e^{3 x}\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^x x \left (-e^x (1-625 x)-e^{2 x} (x-2)+x (x+1)+2 e^x \log (x)\right )}{\left (625 e^x x+x+e^{2 x}+e^x \log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^x x \left (625 e^x x^2+2 x^2-625 e^x x-x-e^x+e^x x \log (x)\right )}{\left (625 e^x x+x+e^{2 x}+e^x \log (x)\right )^2}-\frac {e^x (x-2) x}{625 e^x x+x+e^{2 x}+e^x \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {e^x x^3}{\left (625 e^x x+x+e^{2 x}+e^x \log (x)\right )^2}dx+625 \int \frac {e^{2 x} x^3}{\left (625 e^x x+x+e^{2 x}+e^x \log (x)\right )^2}dx-\int \frac {e^x x^2}{\left (625 e^x x+x+e^{2 x}+e^x \log (x)\right )^2}dx-625 \int \frac {e^{2 x} x^2}{\left (625 e^x x+x+e^{2 x}+e^x \log (x)\right )^2}dx+\int \frac {e^{2 x} x^2 \log (x)}{\left (625 e^x x+x+e^{2 x}+e^x \log (x)\right )^2}dx-\int \frac {e^x x^2}{625 e^x x+x+e^{2 x}+e^x \log (x)}dx-\int \frac {e^{2 x} x}{\left (625 e^x x+x+e^{2 x}+e^x \log (x)\right )^2}dx+2 \int \frac {e^x x}{625 e^x x+x+e^{2 x}+e^x \log (x)}dx\) |
Input:
Int[(E^(3*x)*(2*x - x^2) + E^(2*x)*(-x + 625*x^2) + E^x*(x^2 + x^3) + 2*E^ (2*x)*x*Log[x])/(E^(4*x) + 1250*E^(3*x)*x + x^2 + 1250*E^x*x^2 + E^(2*x)*( 2*x + 390625*x^2) + (2*E^(3*x) + 2*E^x*x + 1250*E^(2*x)*x)*Log[x] + E^(2*x )*Log[x]^2),x]
Output:
$Aborted
Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14
method | result | size |
risch | \(\frac {{\mathrm e}^{x} x^{2}}{{\mathrm e}^{2 x}+{\mathrm e}^{x} \ln \left (x \right )+625 \,{\mathrm e}^{x} x +x}\) | \(25\) |
parallelrisch | \(\frac {{\mathrm e}^{x} x^{2}}{{\mathrm e}^{2 x}+{\mathrm e}^{x} \ln \left (x \right )+625 \,{\mathrm e}^{x} x +x}\) | \(25\) |
Input:
int((2*x*exp(x)^2*ln(x)+(-x^2+2*x)*exp(x)^3+(625*x^2-x)*exp(x)^2+(x^3+x^2) *exp(x))/(exp(x)^2*ln(x)^2+(2*exp(x)^3+1250*x*exp(x)^2+2*exp(x)*x)*ln(x)+e xp(x)^4+1250*x*exp(x)^3+(390625*x^2+2*x)*exp(x)^2+1250*exp(x)*x^2+x^2),x,m ethod=_RETURNVERBOSE)
Output:
exp(x)*x^2/(exp(2*x)+exp(x)*ln(x)+625*exp(x)*x+x)
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {e^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (-x+625 x^2\right )+e^x \left (x^2+x^3\right )+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} \left (2 x+390625 x^2\right )+\left (2 e^{3 x}+2 e^x x+1250 e^{2 x} x\right ) \log (x)+e^{2 x} \log ^2(x)} \, dx=\frac {x^{2} e^{x}}{625 \, x e^{x} + e^{x} \log \left (x\right ) + x + e^{\left (2 \, x\right )}} \] Input:
integrate((2*x*exp(x)^2*log(x)+(-x^2+2*x)*exp(x)^3+(625*x^2-x)*exp(x)^2+(x ^3+x^2)*exp(x))/(exp(x)^2*log(x)^2+(2*exp(x)^3+1250*x*exp(x)^2+2*exp(x)*x) *log(x)+exp(x)^4+1250*x*exp(x)^3+(390625*x^2+2*x)*exp(x)^2+1250*exp(x)*x^2 +x^2),x, algorithm="fricas")
Output:
x^2*e^x/(625*x*e^x + e^x*log(x) + x + e^(2*x))
Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (-x+625 x^2\right )+e^x \left (x^2+x^3\right )+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} \left (2 x+390625 x^2\right )+\left (2 e^{3 x}+2 e^x x+1250 e^{2 x} x\right ) \log (x)+e^{2 x} \log ^2(x)} \, dx=\frac {x^{2} e^{x}}{x + \left (625 x + \log {\left (x \right )}\right ) e^{x} + e^{2 x}} \] Input:
integrate((2*x*exp(x)**2*ln(x)+(-x**2+2*x)*exp(x)**3+(625*x**2-x)*exp(x)** 2+(x**3+x**2)*exp(x))/(exp(x)**2*ln(x)**2+(2*exp(x)**3+1250*x*exp(x)**2+2* exp(x)*x)*ln(x)+exp(x)**4+1250*x*exp(x)**3+(390625*x**2+2*x)*exp(x)**2+125 0*exp(x)*x**2+x**2),x)
Output:
x**2*exp(x)/(x + (625*x + log(x))*exp(x) + exp(2*x))
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {e^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (-x+625 x^2\right )+e^x \left (x^2+x^3\right )+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} \left (2 x+390625 x^2\right )+\left (2 e^{3 x}+2 e^x x+1250 e^{2 x} x\right ) \log (x)+e^{2 x} \log ^2(x)} \, dx=\frac {x^{2} e^{x}}{{\left (625 \, x + \log \left (x\right )\right )} e^{x} + x + e^{\left (2 \, x\right )}} \] Input:
integrate((2*x*exp(x)^2*log(x)+(-x^2+2*x)*exp(x)^3+(625*x^2-x)*exp(x)^2+(x ^3+x^2)*exp(x))/(exp(x)^2*log(x)^2+(2*exp(x)^3+1250*x*exp(x)^2+2*exp(x)*x) *log(x)+exp(x)^4+1250*x*exp(x)^3+(390625*x^2+2*x)*exp(x)^2+1250*exp(x)*x^2 +x^2),x, algorithm="maxima")
Output:
x^2*e^x/((625*x + log(x))*e^x + x + e^(2*x))
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {e^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (-x+625 x^2\right )+e^x \left (x^2+x^3\right )+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} \left (2 x+390625 x^2\right )+\left (2 e^{3 x}+2 e^x x+1250 e^{2 x} x\right ) \log (x)+e^{2 x} \log ^2(x)} \, dx=\frac {x^{2} e^{x}}{625 \, x e^{x} + e^{x} \log \left (x\right ) + x + e^{\left (2 \, x\right )}} \] Input:
integrate((2*x*exp(x)^2*log(x)+(-x^2+2*x)*exp(x)^3+(625*x^2-x)*exp(x)^2+(x ^3+x^2)*exp(x))/(exp(x)^2*log(x)^2+(2*exp(x)^3+1250*x*exp(x)^2+2*exp(x)*x) *log(x)+exp(x)^4+1250*x*exp(x)^3+(390625*x^2+2*x)*exp(x)^2+1250*exp(x)*x^2 +x^2),x, algorithm="giac")
Output:
x^2*e^x/(625*x*e^x + e^x*log(x) + x + e^(2*x))
Time = 0.52 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.45 \[ \int \frac {e^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (-x+625 x^2\right )+e^x \left (x^2+x^3\right )+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} \left (2 x+390625 x^2\right )+\left (2 e^{3 x}+2 e^x x+1250 e^{2 x} x\right ) \log (x)+e^{2 x} \log ^2(x)} \, dx=\frac {{\mathrm {e}}^{2\,x}\,\left (x^4-x^5\right )+x^3\,{\mathrm {e}}^{3\,x}+625\,x^4\,{\mathrm {e}}^{3\,x}+x^4\,{\mathrm {e}}^{4\,x}}{\left (x+{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\ln \left (x\right )+625\,x\,{\mathrm {e}}^x\right )\,\left (x\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^x-x^3\,{\mathrm {e}}^x+625\,x^2\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^{3\,x}\right )} \] Input:
int((exp(3*x)*(2*x - x^2) + exp(x)*(x^2 + x^3) - exp(2*x)*(x - 625*x^2) + 2*x*exp(2*x)*log(x))/(exp(4*x) + log(x)*(2*exp(3*x) + 1250*x*exp(2*x) + 2* x*exp(x)) + exp(2*x)*(2*x + 390625*x^2) + 1250*x*exp(3*x) + 1250*x^2*exp(x ) + x^2 + exp(2*x)*log(x)^2),x)
Output:
(exp(2*x)*(x^4 - x^5) + x^3*exp(3*x) + 625*x^4*exp(3*x) + x^4*exp(4*x))/(( x + exp(2*x) + exp(x)*log(x) + 625*x*exp(x))*(x*exp(2*x) + x^2*exp(x) - x^ 3*exp(x) + 625*x^2*exp(2*x) + x^2*exp(3*x)))
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {e^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (-x+625 x^2\right )+e^x \left (x^2+x^3\right )+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} \left (2 x+390625 x^2\right )+\left (2 e^{3 x}+2 e^x x+1250 e^{2 x} x\right ) \log (x)+e^{2 x} \log ^2(x)} \, dx=\frac {e^{x} x^{2}}{e^{2 x}+e^{x} \mathrm {log}\left (x \right )+625 e^{x} x +x} \] Input:
int((2*x*exp(x)^2*log(x)+(-x^2+2*x)*exp(x)^3+(625*x^2-x)*exp(x)^2+(x^3+x^2 )*exp(x))/(exp(x)^2*log(x)^2+(2*exp(x)^3+1250*x*exp(x)^2+2*exp(x)*x)*log(x )+exp(x)^4+1250*x*exp(x)^3+(390625*x^2+2*x)*exp(x)^2+1250*exp(x)*x^2+x^2), x)
Output:
(e**x*x**2)/(e**(2*x) + e**x*log(x) + 625*e**x*x + x)