\(\int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} (-750 x^3-315 x^4)}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx\) [2802]

Optimal result

Integrand size = 73, antiderivative size = 22 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} e^{-\frac {2 x^4}{\left (x+(5+2 x)^2\right )^2}} \] Output:

15/4/exp(x^4/((5+2*x)^2+x)^2)^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} e^{-\frac {2 x^4}{\left (25+21 x+4 x^2\right )^2}} \] Input:

Integrate[(-750*x^3 - 315*x^4)/(E^((2*x^4)/(625 + 1050*x + 641*x^2 + 168*x 
^3 + 16*x^4))*(15625 + 39375*x + 40575*x^2 + 21861*x^3 + 6492*x^4 + 1008*x 
^5 + 64*x^6)),x]
 

Output:

15/(4*E^((2*x^4)/(25 + 21*x + 4*x^2)^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.

\(\displaystyle \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} \left (-315 x^4-750 x^3\right )}{64 x^6+1008 x^5+6492 x^4+21861 x^3+40575 x^2+39375 x+15625} \, dx\)

\(\Big \downarrow \) 2027

\(\displaystyle \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} (-315 x-750) x^3}{64 x^6+1008 x^5+6492 x^4+21861 x^3+40575 x^2+39375 x+15625}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (-\frac {768 e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} (-315 x-750) x^3}{1681 \sqrt {41} \left (-8 x+\sqrt {41}-21\right )}-\frac {768 e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} (-315 x-750) x^3}{1681 \sqrt {41} \left (8 x+\sqrt {41}+21\right )}-\frac {768 e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} (-315 x-750) x^3}{1681 \left (-8 x+\sqrt {41}-21\right )^2}-\frac {768 e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} (-315 x-750) x^3}{1681 \left (8 x+\sqrt {41}+21\right )^2}-\frac {512 e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} (-315 x-750) x^3}{41 \sqrt {41} \left (-8 x+\sqrt {41}-21\right )^3}-\frac {512 e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} (-315 x-750) x^3}{41 \sqrt {41} \left (8 x+\sqrt {41}+21\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {45 \left (6783+923 \sqrt {41}\right ) \int e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}dx}{13448}-\frac {15 \left (2583+923 \sqrt {41}\right ) \int e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}dx}{13448}-\frac {45 \left (2961+341 \sqrt {41}\right ) \int e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}dx}{6724}-\frac {45 \left (2961-341 \sqrt {41}\right ) \int e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}dx}{6724}+\frac {45 \left (6783-923 \sqrt {41}\right ) \int e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}dx}{13448}-\frac {15 \left (2583-923 \sqrt {41}\right ) \int e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}dx}{13448}-\frac {15}{41} \left (38081-5061 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{\left (-8 x+\sqrt {41}-21\right )^3}dx-\frac {90 \left (54817-8677 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{\left (-8 x+\sqrt {41}-21\right )^2}dx}{1681}+\frac {45 \left (207501-38081 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{\left (-8 x+\sqrt {41}-21\right )^2}dx}{3362}+\frac {135 \left (8677-1337 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{-8 x+\sqrt {41}-21}dx}{1681}-\frac {45 \left (13981-2961 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{-8 x+\sqrt {41}-21}dx}{3362}-\frac {45 \left (38081-5061 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{-8 x+\sqrt {41}-21}dx}{3362}-\frac {180 \left (21+\sqrt {41}\right ) \int e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} x^2dx}{1681}-\frac {180 \left (21-\sqrt {41}\right ) \int e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} x^2dx}{1681}+\frac {7560 \int e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}} x^2dx}{1681}+\frac {15}{41} \left (38081+5061 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{\left (8 x+\sqrt {41}+21\right )^3}dx+\frac {45 \left (207501+38081 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{\left (8 x+\sqrt {41}+21\right )^2}dx}{3362}-\frac {90 \left (54817+8677 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{\left (8 x+\sqrt {41}+21\right )^2}dx}{1681}+\frac {45 \left (38081+5061 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{8 x+\sqrt {41}+21}dx}{3362}+\frac {45 \left (13981+2961 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{8 x+\sqrt {41}+21}dx}{3362}-\frac {135 \left (8677+1337 \sqrt {41}\right ) \int \frac {e^{-\frac {2 x^4}{16 x^4+168 x^3+641 x^2+1050 x+625}}}{8 x+\sqrt {41}+21}dx}{1681}\)

Input:

Int[(-750*x^3 - 315*x^4)/(E^((2*x^4)/(625 + 1050*x + 641*x^2 + 168*x^3 + 1 
6*x^4))*(15625 + 39375*x + 40575*x^2 + 21861*x^3 + 6492*x^4 + 1008*x^5 + 6 
4*x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
risch	\(\frac {15 \,{\mathrm e}^{-\frac {2 x^{4}}{\left (4 x^{2}+21 x +25\right )^{2}}}}{4}\)	\(21\)
gosper	\(\frac {15 \,{\mathrm e}^{-\frac {2 x^{4}}{16 x^{4}+168 x^{3}+641 x^{2}+1050 x +625}}}{4}\)	\(32\)
norman	\(\frac {\left (\frac {9375}{4}+\frac {7875}{2} x +\frac {9615}{4} x^{2}+630 x^{3}+60 x^{4}\right ) {\mathrm e}^{-\frac {2 x^{4}}{16 x^{4}+168 x^{3}+641 x^{2}+1050 x +625}}}{\left (4 x^{2}+21 x +25\right )^{2}}\)	\(63\)
parallelrisch	\(\frac {\left (953164800 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) {\mathrm e}^{-\frac {2 x^{4}}{16 x^{4}+168 x^{3}+641 x^{2}+1050 x +625}}}{15886080 \left (4 x^{2}+21 x +25\right )^{2}}\)	\(64\)
orering	\(-\frac {\left (4 x^{2}+21 x +25\right )^{3} \left (-315 x^{4}-750 x^{3}\right ) {\mathrm e}^{-\frac {2 x^{4}}{16 x^{4}+168 x^{3}+641 x^{2}+1050 x +625}}}{4 x^{3} \left (21 x +50\right ) \left (64 x^{6}+1008 x^{5}+6492 x^{4}+21861 x^{3}+40575 x^{2}+39375 x +15625\right )}\)	\(97\)

Input:

int((-315*x^4-750*x^3)/(64*x^6+1008*x^5+6492*x^4+21861*x^3+40575*x^2+39375 
*x+15625)/exp(x^4/(16*x^4+168*x^3+641*x^2+1050*x+625))^2,x,method=_RETURNV 
ERBOSE)
 

Output:

15/4*exp(-2*x^4/(4*x^2+21*x+25)^2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} \, e^{\left (-\frac {2 \, x^{4}}{16 \, x^{4} + 168 \, x^{3} + 641 \, x^{2} + 1050 \, x + 625}\right )} \] Input:

integrate((-315*x^4-750*x^3)/(64*x^6+1008*x^5+6492*x^4+21861*x^3+40575*x^2 
+39375*x+15625)/exp(x^4/(16*x^4+168*x^3+641*x^2+1050*x+625))^2,x, algorith 
m="fricas")
 

Output:

15/4*e^(-2*x^4/(16*x^4 + 168*x^3 + 641*x^2 + 1050*x + 625))
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15 e^{- \frac {2 x^{4}}{16 x^{4} + 168 x^{3} + 641 x^{2} + 1050 x + 625}}}{4} \] Input:

integrate((-315*x**4-750*x**3)/(64*x**6+1008*x**5+6492*x**4+21861*x**3+405 
75*x**2+39375*x+15625)/exp(x**4/(16*x**4+168*x**3+641*x**2+1050*x+625))**2 
,x)
 

Output:

15*exp(-2*x**4/(16*x**4 + 168*x**3 + 641*x**2 + 1050*x + 625))/4
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (19) = 38\).

Time = 4.55 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.77 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} \, e^{\left (\frac {5061 \, x}{32 \, {\left (16 \, x^{4} + 168 \, x^{3} + 641 \, x^{2} + 1050 \, x + 625\right )}} + \frac {21 \, x}{4 \, {\left (4 \, x^{2} + 21 \, x + 25\right )}} + \frac {8525}{32 \, {\left (16 \, x^{4} + 168 \, x^{3} + 641 \, x^{2} + 1050 \, x + 625\right )}} - \frac {241}{32 \, {\left (4 \, x^{2} + 21 \, x + 25\right )}} - \frac {1}{8}\right )} \] Input:

integrate((-315*x^4-750*x^3)/(64*x^6+1008*x^5+6492*x^4+21861*x^3+40575*x^2 
+39375*x+15625)/exp(x^4/(16*x^4+168*x^3+641*x^2+1050*x+625))^2,x, algorith 
m="maxima")
 

Output:

15/4*e^(5061/32*x/(16*x^4 + 168*x^3 + 641*x^2 + 1050*x + 625) + 21/4*x/(4* 
x^2 + 21*x + 25) + 8525/32/(16*x^4 + 168*x^3 + 641*x^2 + 1050*x + 625) - 2 
41/32/(4*x^2 + 21*x + 25) - 1/8)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} \, e^{\left (-\frac {2 \, x^{4}}{16 \, x^{4} + 168 \, x^{3} + 641 \, x^{2} + 1050 \, x + 625}\right )} \] Input:

integrate((-315*x^4-750*x^3)/(64*x^6+1008*x^5+6492*x^4+21861*x^3+40575*x^2 
+39375*x+15625)/exp(x^4/(16*x^4+168*x^3+641*x^2+1050*x+625))^2,x, algorith 
m="giac")
 

Output:

15/4*e^(-2*x^4/(16*x^4 + 168*x^3 + 641*x^2 + 1050*x + 625))
 

Mupad [B] (verification not implemented)

Time = 2.81 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15\,{\mathrm {e}}^{-\frac {2\,x^4}{16\,x^4+168\,x^3+641\,x^2+1050\,x+625}}}{4} \] Input:

int(-(exp(-(2*x^4)/(1050*x + 641*x^2 + 168*x^3 + 16*x^4 + 625))*(750*x^3 + 
 315*x^4))/(39375*x + 40575*x^2 + 21861*x^3 + 6492*x^4 + 1008*x^5 + 64*x^6 
 + 15625),x)
 

Output:

(15*exp(-(2*x^4)/(1050*x + 641*x^2 + 168*x^3 + 16*x^4 + 625)))/4
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4 e^{\frac {2 x^{4}}{16 x^{4}+168 x^{3}+641 x^{2}+1050 x +625}}} \] Input:

int((-315*x^4-750*x^3)/(64*x^6+1008*x^5+6492*x^4+21861*x^3+40575*x^2+39375 
*x+15625)/exp(x^4/(16*x^4+168*x^3+641*x^2+1050*x+625))^2,x)
 

Output:

15/(4*e**((2*x**4)/(16*x**4 + 168*x**3 + 641*x**2 + 1050*x + 625)))