\(\int \frac {50 x+e^{\frac {1}{625} e^2 (9-150 x+625 x^2)} (-50 x+e^2 (-6 x^2+50 x^3))}{25-50 e^{\frac {1}{625} e^2 (9-150 x+625 x^2)}+25 e^{\frac {2}{625} e^2 (9-150 x+625 x^2)}} \, dx\) [931]

Optimal result

Integrand size = 90, antiderivative size = 23 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=\frac {x^2}{1-e^{e^2 \left (-\frac {3}{25}+x\right )^2}} \] Output:

x^2/(1-exp(exp(1)^2*(x-3/25)^2))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 10.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=-\frac {x^2}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}} \] Input:

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]
 

Output:

-(x^2/(-1 + E^((E^2*(3 - 25*x)^2)/625)))
 

Rubi [F]

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

\(\displaystyle \frac {2}{25} \left (25 e^2 \int \frac {x^3}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}dx+25 e^2 \int \frac {x^3}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}dx-3 e^2 \int \frac {x^2}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}dx-3 e^2 \int \frac {x^2}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}dx-25 \int \frac {x}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}dx\right )\)

Input:

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]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.50 (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	\(-\frac {x^{2}}{{\mathrm e}^{\frac {\left (625 x^{2}-150 x +9\right ) {\mathrm e}^{2}}{625}}-1}\)	\(27\)

Input:

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)
 

Output:

-x^2/(exp(1/625*(25*x-3)^2*exp(2))-1)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=-\frac {x^{2}}{e^{\left (\frac {1}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} - 1} \] Input:

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="fricas")
 

Output:

-x^2/(e^(1/625*(625*x^2 - 150*x + 9)*e^2) - 1)
 

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=- \frac {x^{2}}{e^{\left (x^{2} - \frac {6 x}{25} + \frac {9}{625}\right ) e^{2}} - 1} \] Input:

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)
 

Output:

-x**2/(exp((x**2 - 6*x/25 + 9/625)*exp(2)) - 1)
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=-\frac {x^{2} e^{\left (\frac {6}{25} \, x e^{2}\right )}}{e^{\left (x^{2} e^{2} + \frac {9}{625} \, e^{2}\right )} - e^{\left (\frac {6}{25} \, x e^{2}\right )}} \] Input:

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="maxima")
 

Output:

-x^2*e^(6/25*x*e^2)/(e^(x^2*e^2 + 9/625*e^2) - e^(6/25*x*e^2))
 

Giac [F]

\[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=\int { \frac {2 \, {\left ({\left ({\left (25 \, x^{3} - 3 \, x^{2}\right )} e^{2} - 25 \, x\right )} e^{\left (\frac {1}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} + 25 \, x\right )}}{25 \, {\left (e^{\left (\frac {2}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} - 2 \, e^{\left (\frac {1}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} + 1\right )}} \,d x } \] Input:

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="giac")
 

Output:

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)
 

Mupad [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=-\frac {x^2}{{\mathrm {e}}^{x^2\,{\mathrm {e}}^2}\,{\mathrm {e}}^{\frac {9\,{\mathrm {e}}^2}{625}}\,{\mathrm {e}}^{-\frac {6\,x\,{\mathrm {e}}^2}{25}}-1} \] Input:

int((50*x - exp((exp(2)*(625*x^2 - 150*x + 9))/625)*(50*x + exp(2)*(6*x^2 
- 50*x^3)))/(25*exp((2*exp(2)*(625*x^2 - 150*x + 9))/625) - 50*exp((exp(2) 
*(625*x^2 - 150*x + 9))/625) + 25),x)
 

Output:

-x^2/(exp(x^2*exp(2))*exp((9*exp(2))/625)*exp(-(6*x*exp(2))/25) - 1)
 

Reduce [F]

\[ \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx=2 e^{\frac {9 e^{2}}{625}} \left (\int \frac {e^{e^{2} x^{2}+\frac {6}{25} e^{2} x} x^{3}}{e^{2 e^{2} x^{2}+\frac {18}{625} e^{2}}-2 e^{e^{2} x^{2}+\frac {6}{25} e^{2} x +\frac {9}{625} e^{2}}+e^{\frac {12 e^{2} x}{25}}}d x \right ) e^{2}-\frac {6 e^{\frac {9 e^{2}}{625}} \left (\int \frac {e^{e^{2} x^{2}+\frac {6}{25} e^{2} x} x^{2}}{e^{2 e^{2} x^{2}+\frac {18}{625} e^{2}}-2 e^{e^{2} x^{2}+\frac {6}{25} e^{2} x +\frac {9}{625} e^{2}}+e^{\frac {12 e^{2} x}{25}}}d x \right ) e^{2}}{25}-2 e^{\frac {9 e^{2}}{625}} \left (\int \frac {e^{e^{2} x^{2}+\frac {6}{25} e^{2} x} x}{e^{2 e^{2} x^{2}+\frac {18}{625} e^{2}}-2 e^{e^{2} x^{2}+\frac {6}{25} e^{2} x +\frac {9}{625} e^{2}}+e^{\frac {12 e^{2} x}{25}}}d x \right )+2 \left (\int \frac {e^{\frac {12 e^{2} x}{25}} x}{e^{2 e^{2} x^{2}+\frac {18}{625} e^{2}}-2 e^{e^{2} x^{2}+\frac {6}{25} e^{2} x +\frac {9}{625} e^{2}}+e^{\frac {12 e^{2} x}{25}}}d x \right ) \] Input:

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)
 

Output:

(2*(25*e**((9*e**2)/625)*int((e**((25*e**2*x**2 + 6*e**2*x)/25)*x**3)/(e** 
((1250*e**2*x**2 + 18*e**2)/625) - 2*e**((625*e**2*x**2 + 150*e**2*x + 9*e 
**2)/625) + e**((12*e**2*x)/25)),x)*e**2 - 3*e**((9*e**2)/625)*int((e**((2 
5*e**2*x**2 + 6*e**2*x)/25)*x**2)/(e**((1250*e**2*x**2 + 18*e**2)/625) - 2 
*e**((625*e**2*x**2 + 150*e**2*x + 9*e**2)/625) + e**((12*e**2*x)/25)),x)* 
e**2 - 25*e**((9*e**2)/625)*int((e**((25*e**2*x**2 + 6*e**2*x)/25)*x)/(e** 
((1250*e**2*x**2 + 18*e**2)/625) - 2*e**((625*e**2*x**2 + 150*e**2*x + 9*e 
**2)/625) + e**((12*e**2*x)/25)),x) + 25*int((e**((12*e**2*x)/25)*x)/(e**( 
(1250*e**2*x**2 + 18*e**2)/625) - 2*e**((625*e**2*x**2 + 150*e**2*x + 9*e* 
*2)/625) + e**((12*e**2*x)/25)),x)))/25