\(\int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} (-281250-93750 x+e^{x^2} (-562500 x^2-375000 x^3-62500 x^4))}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} (1687500 x^3+286875 x^4)+e^{e^{x^2}} (6750000 x^3+2295000 x^4+195075 x^5)} \, dx\) [2340]

Optimal result

Integrand size = 133, antiderivative size = 31 \[ \int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} \left (-281250-93750 x+e^{x^2} \left (-562500 x^2-375000 x^3-62500 x^4\right )\right )}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} \left (1687500 x^3+286875 x^4\right )+e^{e^{x^2}} \left (6750000 x^3+2295000 x^4+195075 x^5\right )} \, dx=\frac {(3+x)^2}{x^2 \left (-\frac {126 x}{25}+3 \left (-4-e^{e^{x^2}}+x\right )\right )^2} \] Output:

(3+x)^2/(-51/25*x-12-3*exp(exp(x^2)))^2/x^2
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} \left (-281250-93750 x+e^{x^2} \left (-562500 x^2-375000 x^3-62500 x^4\right )\right )}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} \left (1687500 x^3+286875 x^4\right )+e^{e^{x^2}} \left (6750000 x^3+2295000 x^4+195075 x^5\right )} \, dx=\frac {625 (3+x)^2}{9 x^2 \left (100+25 e^{e^{x^2}}+17 x\right )^2} \] Input:

Integrate[(-1125000 - 757500*x - 191250*x^2 - 21250*x^3 + E^E^x^2*(-281250 
 - 93750*x + E^x^2*(-562500*x^2 - 375000*x^3 - 62500*x^4)))/(9000000*x^3 + 
 140625*E^(3*E^x^2)*x^3 + 4590000*x^4 + 780300*x^5 + 44217*x^6 + E^(2*E^x^ 
2)*(1687500*x^3 + 286875*x^4) + E^E^x^2*(6750000*x^3 + 2295000*x^4 + 19507 
5*x^5)),x]
 

Output:

(625*(3 + x)^2)/(9*x^2*(100 + 25*E^E^x^2 + 17*x)^2)
 

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.

\(\displaystyle \int \frac {-21250 x^3-191250 x^2+e^{e^{x^2}} \left (e^{x^2} \left (-62500 x^4-375000 x^3-562500 x^2\right )-93750 x-281250\right )-757500 x-1125000}{44217 x^6+780300 x^5+4590000 x^4+9000000 x^3+140625 e^{3 e^{x^2}} x^3+e^{2 e^{x^2}} \left (286875 x^4+1687500 x^3\right )+e^{e^{x^2}} \left (195075 x^5+2295000 x^4+6750000 x^3\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1250 (x+3) \left (-50 e^{x^2+e^{x^2}} (x+3) x^2-17 x^2-75 e^{e^{x^2}}-102 x-300\right )}{9 x^3 \left (25 e^{e^{x^2}}+17 x+100\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1250}{9} \int -\frac {(x+3) \left (50 e^{x^2+e^{x^2}} (x+3) x^2+17 x^2+102 x+75 e^{e^{x^2}}+300\right )}{x^3 \left (17 x+25 e^{e^{x^2}}+100\right )^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1250}{9} \int \frac {(x+3) \left (50 e^{x^2+e^{x^2}} (x+3) x^2+17 x^2+102 x+75 e^{e^{x^2}}+300\right )}{x^3 \left (17 x+25 e^{e^{x^2}}+100\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1250}{9} \int \left (\frac {50 e^{x^2+e^{x^2}} (x+3)^2}{x \left (17 x+25 e^{e^{x^2}}+100\right )^3}+\frac {17 (x+3)}{x \left (17 x+25 e^{e^{x^2}}+100\right )^3}+\frac {102 (x+3)}{x^2 \left (17 x+25 e^{e^{x^2}}+100\right )^3}+\frac {75 e^{e^{x^2}} (x+3)}{x^3 \left (17 x+25 e^{e^{x^2}}+100\right )^3}+\frac {300 (x+3)}{x^3 \left (17 x+25 e^{e^{x^2}}+100\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1250}{9} \left (17 \int \frac {1}{\left (17 x+25 e^{e^{x^2}}+100\right )^3}dx+300 \int \frac {e^{x^2+e^{x^2}}}{\left (17 x+25 e^{e^{x^2}}+100\right )^3}dx+606 \int \frac {1}{x^2 \left (17 x+25 e^{e^{x^2}}+100\right )^3}dx+75 \int \frac {e^{e^{x^2}}}{x^2 \left (17 x+25 e^{e^{x^2}}+100\right )^3}dx+153 \int \frac {1}{x \left (17 x+25 e^{e^{x^2}}+100\right )^3}dx+450 \int \frac {e^{x^2+e^{x^2}}}{x \left (17 x+25 e^{e^{x^2}}+100\right )^3}dx+50 \int \frac {e^{x^2+e^{x^2}} x}{\left (17 x+25 e^{e^{x^2}}+100\right )^3}dx+900 \int \frac {1}{x^3 \left (17 x+25 e^{e^{x^2}}+100\right )^3}dx+225 \int \frac {e^{e^{x^2}}}{x^3 \left (17 x+25 e^{e^{x^2}}+100\right )^3}dx\right )\)

Input:

Int[(-1125000 - 757500*x - 191250*x^2 - 21250*x^3 + E^E^x^2*(-281250 - 937 
50*x + E^x^2*(-562500*x^2 - 375000*x^3 - 62500*x^4)))/(9000000*x^3 + 14062 
5*E^(3*E^x^2)*x^3 + 4590000*x^4 + 780300*x^5 + 44217*x^6 + E^(2*E^x^2)*(16 
87500*x^3 + 286875*x^4) + E^E^x^2*(6750000*x^3 + 2295000*x^4 + 195075*x^5) 
),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90

method	result	size
risch	\(\frac {\frac {625}{9} x^{2}+\frac {1250}{3} x +625}{x^{2} \left (25 \,{\mathrm e}^{{\mathrm e}^{x^{2}}}+17 x +100\right )^{2}}\)	\(28\)
parallelrisch	\(\frac {390625 x^{2}+2343750 x +3515625}{5625 x^{2} \left (289 x^{2}+850 \,{\mathrm e}^{{\mathrm e}^{x^{2}}} x +625 \,{\mathrm e}^{2 \,{\mathrm e}^{x^{2}}}+3400 x +5000 \,{\mathrm e}^{{\mathrm e}^{x^{2}}}+10000\right )}\)	\(52\)

Input:

int((((-62500*x^4-375000*x^3-562500*x^2)*exp(x^2)-93750*x-281250)*exp(exp( 
x^2))-21250*x^3-191250*x^2-757500*x-1125000)/(140625*x^3*exp(exp(x^2))^3+( 
286875*x^4+1687500*x^3)*exp(exp(x^2))^2+(195075*x^5+2295000*x^4+6750000*x^ 
3)*exp(exp(x^2))+44217*x^6+780300*x^5+4590000*x^4+9000000*x^3),x,method=_R 
ETURNVERBOSE)
 

Output:

625/9*(x^2+6*x+9)/x^2/(25*exp(exp(x^2))+17*x+100)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).

Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} \left (-281250-93750 x+e^{x^2} \left (-562500 x^2-375000 x^3-62500 x^4\right )\right )}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} \left (1687500 x^3+286875 x^4\right )+e^{e^{x^2}} \left (6750000 x^3+2295000 x^4+195075 x^5\right )} \, dx=\frac {625 \, {\left (x^{2} + 6 \, x + 9\right )}}{9 \, {\left (289 \, x^{4} + 3400 \, x^{3} + 625 \, x^{2} e^{\left (2 \, e^{\left (x^{2}\right )}\right )} + 10000 \, x^{2} + 50 \, {\left (17 \, x^{3} + 100 \, x^{2}\right )} e^{\left (e^{\left (x^{2}\right )}\right )}\right )}} \] Input:

integrate((((-62500*x^4-375000*x^3-562500*x^2)*exp(x^2)-93750*x-281250)*ex 
p(exp(x^2))-21250*x^3-191250*x^2-757500*x-1125000)/(140625*x^3*exp(exp(x^2 
))^3+(286875*x^4+1687500*x^3)*exp(exp(x^2))^2+(195075*x^5+2295000*x^4+6750 
000*x^3)*exp(exp(x^2))+44217*x^6+780300*x^5+4590000*x^4+9000000*x^3),x, al 
gorithm="fricas")
 

Output:

625/9*(x^2 + 6*x + 9)/(289*x^4 + 3400*x^3 + 625*x^2*e^(2*e^(x^2)) + 10000* 
x^2 + 50*(17*x^3 + 100*x^2)*e^(e^(x^2)))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).

Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} \left (-281250-93750 x+e^{x^2} \left (-562500 x^2-375000 x^3-62500 x^4\right )\right )}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} \left (1687500 x^3+286875 x^4\right )+e^{e^{x^2}} \left (6750000 x^3+2295000 x^4+195075 x^5\right )} \, dx=\frac {625 x^{2} + 3750 x + 5625}{2601 x^{4} + 30600 x^{3} + 5625 x^{2} e^{2 e^{x^{2}}} + 90000 x^{2} + \left (7650 x^{3} + 45000 x^{2}\right ) e^{e^{x^{2}}}} \] Input:

integrate((((-62500*x**4-375000*x**3-562500*x**2)*exp(x**2)-93750*x-281250 
)*exp(exp(x**2))-21250*x**3-191250*x**2-757500*x-1125000)/(140625*x**3*exp 
(exp(x**2))**3+(286875*x**4+1687500*x**3)*exp(exp(x**2))**2+(195075*x**5+2 
295000*x**4+6750000*x**3)*exp(exp(x**2))+44217*x**6+780300*x**5+4590000*x* 
*4+9000000*x**3),x)
 

Output:

(625*x**2 + 3750*x + 5625)/(2601*x**4 + 30600*x**3 + 5625*x**2*exp(2*exp(x 
**2)) + 90000*x**2 + (7650*x**3 + 45000*x**2)*exp(exp(x**2)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} \left (-281250-93750 x+e^{x^2} \left (-562500 x^2-375000 x^3-62500 x^4\right )\right )}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} \left (1687500 x^3+286875 x^4\right )+e^{e^{x^2}} \left (6750000 x^3+2295000 x^4+195075 x^5\right )} \, dx=\frac {625 \, {\left (x^{2} + 6 \, x + 9\right )}}{9 \, {\left (289 \, x^{4} + 3400 \, x^{3} + 625 \, x^{2} e^{\left (2 \, e^{\left (x^{2}\right )}\right )} + 10000 \, x^{2} + 50 \, {\left (17 \, x^{3} + 100 \, x^{2}\right )} e^{\left (e^{\left (x^{2}\right )}\right )}\right )}} \] Input:

integrate((((-62500*x^4-375000*x^3-562500*x^2)*exp(x^2)-93750*x-281250)*ex 
p(exp(x^2))-21250*x^3-191250*x^2-757500*x-1125000)/(140625*x^3*exp(exp(x^2 
))^3+(286875*x^4+1687500*x^3)*exp(exp(x^2))^2+(195075*x^5+2295000*x^4+6750 
000*x^3)*exp(exp(x^2))+44217*x^6+780300*x^5+4590000*x^4+9000000*x^3),x, al 
gorithm="maxima")
 

Output:

625/9*(x^2 + 6*x + 9)/(289*x^4 + 3400*x^3 + 625*x^2*e^(2*e^(x^2)) + 10000* 
x^2 + 50*(17*x^3 + 100*x^2)*e^(e^(x^2)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).

Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.94 \[ \int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} \left (-281250-93750 x+e^{x^2} \left (-562500 x^2-375000 x^3-62500 x^4\right )\right )}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} \left (1687500 x^3+286875 x^4\right )+e^{e^{x^2}} \left (6750000 x^3+2295000 x^4+195075 x^5\right )} \, dx=\frac {625 \, {\left (x^{2} + 6 \, x + 9\right )}}{9 \, {\left (289 \, x^{4} + 850 \, x^{3} e^{\left (e^{\left (x^{2}\right )}\right )} + 3400 \, x^{3} + 625 \, x^{2} e^{\left (2 \, e^{\left (x^{2}\right )}\right )} + 5000 \, x^{2} e^{\left (e^{\left (x^{2}\right )}\right )} + 10000 \, x^{2}\right )}} \] Input:

integrate((((-62500*x^4-375000*x^3-562500*x^2)*exp(x^2)-93750*x-281250)*ex 
p(exp(x^2))-21250*x^3-191250*x^2-757500*x-1125000)/(140625*x^3*exp(exp(x^2 
))^3+(286875*x^4+1687500*x^3)*exp(exp(x^2))^2+(195075*x^5+2295000*x^4+6750 
000*x^3)*exp(exp(x^2))+44217*x^6+780300*x^5+4590000*x^4+9000000*x^3),x, al 
gorithm="giac")
 

Output:

625/9*(x^2 + 6*x + 9)/(289*x^4 + 850*x^3*e^(e^(x^2)) + 3400*x^3 + 625*x^2* 
e^(2*e^(x^2)) + 5000*x^2*e^(e^(x^2)) + 10000*x^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} \left (-281250-93750 x+e^{x^2} \left (-562500 x^2-375000 x^3-62500 x^4\right )\right )}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} \left (1687500 x^3+286875 x^4\right )+e^{e^{x^2}} \left (6750000 x^3+2295000 x^4+195075 x^5\right )} \, dx=\int -\frac {757500\,x+191250\,x^2+21250\,x^3+{\mathrm {e}}^{{\mathrm {e}}^{x^2}}\,\left (93750\,x+{\mathrm {e}}^{x^2}\,\left (62500\,x^4+375000\,x^3+562500\,x^2\right )+281250\right )+1125000}{{\mathrm {e}}^{{\mathrm {e}}^{x^2}}\,\left (195075\,x^5+2295000\,x^4+6750000\,x^3\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^{x^2}}\,\left (286875\,x^4+1687500\,x^3\right )+140625\,x^3\,{\mathrm {e}}^{3\,{\mathrm {e}}^{x^2}}+9000000\,x^3+4590000\,x^4+780300\,x^5+44217\,x^6} \,d x \] Input:

int(-(757500*x + 191250*x^2 + 21250*x^3 + exp(exp(x^2))*(93750*x + exp(x^2 
)*(562500*x^2 + 375000*x^3 + 62500*x^4) + 281250) + 1125000)/(exp(exp(x^2) 
)*(6750000*x^3 + 2295000*x^4 + 195075*x^5) + exp(2*exp(x^2))*(1687500*x^3 
+ 286875*x^4) + 140625*x^3*exp(3*exp(x^2)) + 9000000*x^3 + 4590000*x^4 + 7 
80300*x^5 + 44217*x^6),x)
 

Output:

int(-(757500*x + 191250*x^2 + 21250*x^3 + exp(exp(x^2))*(93750*x + exp(x^2 
)*(562500*x^2 + 375000*x^3 + 62500*x^4) + 281250) + 1125000)/(exp(exp(x^2) 
)*(6750000*x^3 + 2295000*x^4 + 195075*x^5) + exp(2*exp(x^2))*(1687500*x^3 
+ 286875*x^4) + 140625*x^3*exp(3*exp(x^2)) + 9000000*x^3 + 4590000*x^4 + 7 
80300*x^5 + 44217*x^6), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {-1125000-757500 x-191250 x^2-21250 x^3+e^{e^{x^2}} \left (-281250-93750 x+e^{x^2} \left (-562500 x^2-375000 x^3-62500 x^4\right )\right )}{9000000 x^3+140625 e^{3 e^{x^2}} x^3+4590000 x^4+780300 x^5+44217 x^6+e^{2 e^{x^2}} \left (1687500 x^3+286875 x^4\right )+e^{e^{x^2}} \left (6750000 x^3+2295000 x^4+195075 x^5\right )} \, dx=\frac {\frac {625}{9} x^{2}+\frac {1250}{3} x +625}{x^{2} \left (625 e^{2 e^{x^{2}}}+850 e^{e^{x^{2}}} x +5000 e^{e^{x^{2}}}+289 x^{2}+3400 x +10000\right )} \] Input:

int((((-62500*x^4-375000*x^3-562500*x^2)*exp(x^2)-93750*x-281250)*exp(exp( 
x^2))-21250*x^3-191250*x^2-757500*x-1125000)/(140625*x^3*exp(exp(x^2))^3+( 
286875*x^4+1687500*x^3)*exp(exp(x^2))^2+(195075*x^5+2295000*x^4+6750000*x^ 
3)*exp(exp(x^2))+44217*x^6+780300*x^5+4590000*x^4+9000000*x^3),x)
 

Output:

(625*(x**2 + 6*x + 9))/(9*x**2*(625*e**(2*e**(x**2)) + 850*e**(e**(x**2))* 
x + 5000*e**(e**(x**2)) + 289*x**2 + 3400*x + 10000))