Integrand size = 90, antiderivative size = 23 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=\frac {x^2}{1-e^{e^2 \left (-\frac {3}{25}+x\right )^2}} \]
Time = 10.37 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=-\frac {x^2}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}} \]
Integrate[(50*x + E^((E^2*(9 - 150*x + 625*x^2))/625)*(-50*x + E^2*(-6*x^2 + 50*x^3)))/(25 - 50*E^((E^2*(9 - 150*x + 625*x^2))/625) + 25*E^((2*E^2*( 9 - 150*x + 625*x^2))/625)),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.
\(\displaystyle \int \frac {e^{\frac {1}{625} e^2 \left (625 x^2-150 x+9\right )} \left (e^2 \left (50 x^3-6 x^2\right )-50 x\right )+50 x}{-50 e^{\frac {1}{625} e^2 \left (625 x^2-150 x+9\right )}+25 e^{\frac {2}{625} e^2 \left (625 x^2-150 x+9\right )}+25} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {1}{625} e^2 \left (625 x^2-150 x+9\right )} \left (e^2 \left (50 x^3-6 x^2\right )-50 x\right )+50 x}{25 \left (1-e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{25} \int \frac {2 \left (25 x-e^{\frac {1}{625} e^2 \left (625 x^2-150 x+9\right )} \left (25 x+e^2 \left (3 x^2-25 x^3\right )\right )\right )}{\left (1-e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{25} \int \frac {25 x-e^{\frac {1}{625} e^2 \left (625 x^2-150 x+9\right )} \left (25 x+e^2 \left (3 x^2-25 x^3\right )\right )}{\left (1-e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {2}{25} \int \left (\frac {e^2 (25 x-3) x^2}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}+\frac {\left (25 e^2 x^2-3 e^2 x-25\right ) x}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{25} \left (25 e^2 \int \frac {x^3}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}dx+25 e^2 \int \frac {x^3}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}dx-3 e^2 \int \frac {x^2}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}dx-3 e^2 \int \frac {x^2}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}dx-25 \int \frac {x}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}dx\right )\) |
Int[(50*x + E^((E^2*(9 - 150*x + 625*x^2))/625)*(-50*x + E^2*(-6*x^2 + 50* x^3)))/(25 - 50*E^((E^2*(9 - 150*x + 625*x^2))/625) + 25*E^((2*E^2*(9 - 15 0*x + 625*x^2))/625)),x]
3.10.31.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 1.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-\frac {x^{2}}{{\mathrm e}^{\frac {\left (25 x -3\right )^{2} {\mathrm e}^{2}}{625}}-1}\) | \(22\) |
norman | \(-\frac {x^{2}}{{\mathrm e}^{\frac {\left (625 x^{2}-150 x +9\right ) {\mathrm e}^{2}}{625}}-1}\) | \(27\) |
parallelrisch | \(-\frac {x^{2}}{{\mathrm e}^{\frac {\left (625 x^{2}-150 x +9\right ) {\mathrm e}^{2}}{625}}-1}\) | \(27\) |
int((((50*x^3-6*x^2)*exp(1)^2-50*x)*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)+ 50*x)/(25*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)^2-50*exp(1/625*(625*x^2-15 0*x+9)*exp(1)^2)+25),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=-\frac {x^{2}}{e^{\left (\frac {1}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} - 1} \]
integrate((((50*x^3-6*x^2)*exp(1)^2-50*x)*exp(1/625*(625*x^2-150*x+9)*exp( 1)^2)+50*x)/(25*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)^2-50*exp(1/625*(625* x^2-150*x+9)*exp(1)^2)+25),x, algorithm=\
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=- \frac {x^{2}}{e^{\left (x^{2} - \frac {6 x}{25} + \frac {9}{625}\right ) e^{2}} - 1} \]
integrate((((50*x**3-6*x**2)*exp(1)**2-50*x)*exp(1/625*(625*x**2-150*x+9)* exp(1)**2)+50*x)/(25*exp(1/625*(625*x**2-150*x+9)*exp(1)**2)**2-50*exp(1/6 25*(625*x**2-150*x+9)*exp(1)**2)+25),x)
Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=-\frac {x^{2} e^{\left (\frac {6}{25} \, x e^{2}\right )}}{e^{\left (x^{2} e^{2} + \frac {9}{625} \, e^{2}\right )} - e^{\left (\frac {6}{25} \, x e^{2}\right )}} \]
integrate((((50*x^3-6*x^2)*exp(1)^2-50*x)*exp(1/625*(625*x^2-150*x+9)*exp( 1)^2)+50*x)/(25*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)^2-50*exp(1/625*(625* x^2-150*x+9)*exp(1)^2)+25),x, algorithm=\
\[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=\int { \frac {2 \, {\left ({\left ({\left (25 \, x^{3} - 3 \, x^{2}\right )} e^{2} - 25 \, x\right )} e^{\left (\frac {1}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} + 25 \, x\right )}}{25 \, {\left (e^{\left (\frac {2}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} - 2 \, e^{\left (\frac {1}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} + 1\right )}} \,d x } \]
integrate((((50*x^3-6*x^2)*exp(1)^2-50*x)*exp(1/625*(625*x^2-150*x+9)*exp( 1)^2)+50*x)/(25*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)^2-50*exp(1/625*(625* x^2-150*x+9)*exp(1)^2)+25),x, algorithm=\
integrate(2/25*(((25*x^3 - 3*x^2)*e^2 - 25*x)*e^(1/625*(625*x^2 - 150*x + 9)*e^2) + 25*x)/(e^(2/625*(625*x^2 - 150*x + 9)*e^2) - 2*e^(1/625*(625*x^2 - 150*x + 9)*e^2) + 1), x)
Time = 0.79 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=-\frac {x^2}{{\mathrm {e}}^{x^2\,{\mathrm {e}}^2}\,{\mathrm {e}}^{\frac {9\,{\mathrm {e}}^2}{625}}\,{\mathrm {e}}^{-\frac {6\,x\,{\mathrm {e}}^2}{25}}-1} \]