\(\int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} (-15+16 x-16 x^2))}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} (-24 x-16 x^3)} \, dx\) [2092]

Optimal result

Integrand size = 123, antiderivative size = 25 \[ \int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} \left (-24 x-16 x^3\right )} \, dx=\frac {-1+e^x}{-\frac {3}{e^{x/4}-2 x}+x} \] Output:

(-1+exp(x))/(x-3/(exp(1/4*x)-2*x))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 7.94 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} \left (-24 x-16 x^3\right )} \, dx=\frac {\left (-1+e^x\right ) \left (e^{x/4}-2 x\right )}{-3+e^{x/4} x-2 x^2} \] Input:

Integrate[(-24 + 4*E^(x/2) + E^(x/4)*(3 - 16*x) + 16*x^2 + E^x*(24 + 24*x 
- 16*x^2 + 16*x^3 + E^(x/2)*(-4 + 4*x) + E^(x/4)*(-15 + 16*x - 16*x^2)))/( 
36 + 48*x^2 + 4*E^(x/2)*x^2 + 16*x^4 + E^(x/4)*(-24*x - 16*x^3)),x]
 

Output:

((-1 + E^x)*(E^(x/4) - 2*x))/(-3 + E^(x/4)*x - 2*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.

\(\displaystyle \int \frac {16 x^2+e^x \left (16 x^3-16 x^2+e^{x/4} \left (-16 x^2+16 x-15\right )+24 x+e^{x/2} (4 x-4)+24\right )+4 e^{x/2}+e^{x/4} (3-16 x)-24}{16 x^4+e^{x/4} \left (-16 x^3-24 x\right )+4 e^{x/2} x^2+48 x^2+36} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {16 x^2+e^x \left (16 x^3-16 x^2+e^{x/4} \left (-16 x^2+16 x-15\right )+24 x+e^{x/2} (4 x-4)+24\right )+4 e^{x/2}+e^{x/4} (3-16 x)-24}{4 \left (2 x^2-e^{x/4} x+3\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int -\frac {-16 x^2-4 e^{x/2}-e^{x/4} (3-16 x)-e^x \left (16 x^3-16 x^2+24 x-4 e^{x/2} (1-x)-e^{x/4} \left (16 x^2-16 x+15\right )+24\right )+24}{\left (2 x^2-e^{x/4} x+3\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{4} \int \frac {-16 x^2-4 e^{x/2}-e^{x/4} (3-16 x)-e^x \left (16 x^3-16 x^2+24 x-4 e^{x/2} (1-x)-e^{x/4} \left (16 x^2-16 x+15\right )+24\right )+24}{\left (2 x^2-e^{x/4} x+3\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (-\frac {4 e^x (x-1)}{x^2}-\frac {3 e^{3 x/4} (3 x-8)}{x^3}-\frac {6 e^{x/2} \left (2 x^3-4 x^2+3 x-18\right )}{x^4}-\frac {3 e^{x/4} \left (4 x^5+12 x^3-96 x^2+9 x-144\right )}{x^5}-\frac {4 \left (24 x^6-107 x^4-486 x^2-405\right )}{x^6}-\frac {3 \left (16 x^9-128 x^8+96 x^7+215 x^5+1720 x^4+216 x^3+3456 x^2+81 x+1944\right )}{x^6 \left (2 x^2-e^{x/4} x+3\right )}+\frac {3 \left (32 x^{11}-128 x^{10}+240 x^9-576 x^8+718 x^7-568 x^6+1077 x^5+852 x^4+810 x^3+1944 x^2+243 x+972\right )}{x^6 \left (2 x^2-e^{x/4} x+3\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \left (1704 \int \frac {1}{\left (-2 x^2+e^{x/4} x-3\right )^2}dx-2556 \int \frac {1}{x^2 \left (2 x^2-e^{x/4} x+3\right )^2}dx-3231 \int \frac {1}{x \left (2 x^2-e^{x/4} x+3\right )^2}dx-2154 \int \frac {x}{\left (2 x^2-e^{x/4} x+3\right )^2}dx+1728 \int \frac {x^2}{\left (2 x^2-e^{x/4} x+3\right )^2}dx+5160 \int \frac {1}{x^2 \left (2 x^2-e^{x/4} x+3\right )}dx+645 \int \frac {1}{x \left (2 x^2-e^{x/4} x+3\right )}dx+288 \int \frac {x}{2 x^2-e^{x/4} x+3}dx-384 \int \frac {x^2}{2 x^2-e^{x/4} x+3}dx-2916 \int \frac {1}{x^6 \left (2 x^2-e^{x/4} x+3\right )^2}dx+5832 \int \frac {1}{x^6 \left (2 x^2-e^{x/4} x+3\right )}dx-729 \int \frac {1}{x^5 \left (2 x^2-e^{x/4} x+3\right )^2}dx-96 \int \frac {x^5}{\left (2 x^2-e^{x/4} x+3\right )^2}dx+243 \int \frac {1}{x^5 \left (2 x^2-e^{x/4} x+3\right )}dx-5832 \int \frac {1}{x^4 \left (2 x^2-e^{x/4} x+3\right )^2}dx+384 \int \frac {x^4}{\left (2 x^2-e^{x/4} x+3\right )^2}dx+10368 \int \frac {1}{x^4 \left (2 x^2-e^{x/4} x+3\right )}dx-2430 \int \frac {1}{x^3 \left (2 x^2-e^{x/4} x+3\right )^2}dx-720 \int \frac {x^3}{\left (2 x^2-e^{x/4} x+3\right )^2}dx+648 \int \frac {1}{x^3 \left (2 x^2-e^{x/4} x+3\right )}dx+48 \int \frac {x^3}{2 x^2-e^{x/4} x+3}dx+\frac {324}{x^5}+\frac {108 e^{x/4}}{x^4}+\frac {36 e^{x/2}}{x^3}+\frac {648}{x^3}+\frac {144 e^{x/4}}{x^2}+\frac {12 e^{3 x/4}}{x^2}+96 x+48 e^{x/4}+\frac {24 e^{x/2}}{x}+\frac {4 e^x}{x}+\frac {428}{x}\right )\)

Input:

Int[(-24 + 4*E^(x/2) + E^(x/4)*(3 - 16*x) + 16*x^2 + E^x*(24 + 24*x - 16*x 
^2 + 16*x^3 + E^(x/2)*(-4 + 4*x) + E^(x/4)*(-15 + 16*x - 16*x^2)))/(36 + 4 
8*x^2 + 4*E^(x/2)*x^2 + 16*x^4 + E^(x/4)*(-24*x - 16*x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68

method	result	size
parallelrisch	\(-\frac {-8 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} {\mathrm e}^{\frac {x}{4}}+8 x -4 \,{\mathrm e}^{\frac {x}{4}}}{4 \left (2 x^{2}-{\mathrm e}^{\frac {x}{4}} x +3\right )}\)	\(42\)
risch	\(\frac {x^{3} {\mathrm e}^{\frac {5 x}{4}}-2 \,{\mathrm e}^{x} x^{4}-{\mathrm e}^{\frac {x}{4}} x^{3}+2 x^{4}}{x^{3} \left ({\mathrm e}^{\frac {x}{4}} x -2 x^{2}-3\right )}\)	\(50\)

Input:

int((((-4+4*x)*exp(1/4*x)^2+(-16*x^2+16*x-15)*exp(1/4*x)+16*x^3-16*x^2+24* 
x+24)*exp(x)+4*exp(1/4*x)^2+(3-16*x)*exp(1/4*x)+16*x^2-24)/(4*x^2*exp(1/4* 
x)^2+(-16*x^3-24*x)*exp(1/4*x)+16*x^4+48*x^2+36),x,method=_RETURNVERBOSE)
 

Output:

-1/4*(-8*exp(x)*x+4*exp(x)*exp(1/4*x)+8*x-4*exp(1/4*x))/(2*x^2-exp(1/4*x)* 
x+3)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} \left (-24 x-16 x^3\right )} \, dx=\frac {2 \, x e^{x} - 2 \, x - e^{\left (\frac {5}{4} \, x\right )} + e^{\left (\frac {1}{4} \, x\right )}}{2 \, x^{2} - x e^{\left (\frac {1}{4} \, x\right )} + 3} \] Input:

integrate((((-4+4*x)*exp(1/4*x)^2+(-16*x^2+16*x-15)*exp(1/4*x)+16*x^3-16*x 
^2+24*x+24)*exp(x)+4*exp(1/4*x)^2+(3-16*x)*exp(1/4*x)+16*x^2-24)/(4*x^2*ex 
p(1/4*x)^2+(-16*x^3-24*x)*exp(1/4*x)+16*x^4+48*x^2+36),x, algorithm="frica 
s")
 

Output:

(2*x*e^x - 2*x - e^(5/4*x) + e^(1/4*x))/(2*x^2 - x*e^(1/4*x) + 3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.64 \[ \int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} \left (-24 x-16 x^3\right )} \, dx=24 x + \frac {48 x^{8} + 288 x^{6} + 645 x^{4} + 648 x^{2} + 243}{- 2 x^{7} + x^{6} e^{\frac {x}{4}} - 3 x^{5}} + \frac {107 x^{4} + 162 x^{2} + 81}{x^{5}} + \frac {x^{9} e^{x} + 3 x^{8} e^{\frac {3 x}{4}} + \left (6 x^{9} + 9 x^{7}\right ) e^{\frac {x}{2}} + \left (12 x^{10} + 36 x^{8} + 27 x^{6}\right ) e^{\frac {x}{4}}}{x^{10}} \] Input:

integrate((((-4+4*x)*exp(1/4*x)**2+(-16*x**2+16*x-15)*exp(1/4*x)+16*x**3-1 
6*x**2+24*x+24)*exp(x)+4*exp(1/4*x)**2+(3-16*x)*exp(1/4*x)+16*x**2-24)/(4* 
x**2*exp(1/4*x)**2+(-16*x**3-24*x)*exp(1/4*x)+16*x**4+48*x**2+36),x)
 

Output:

24*x + (48*x**8 + 288*x**6 + 645*x**4 + 648*x**2 + 243)/(-2*x**7 + x**6*ex 
p(x/4) - 3*x**5) + (107*x**4 + 162*x**2 + 81)/x**5 + (x**9*exp(x) + 3*x**8 
*exp(3*x/4) + (6*x**9 + 9*x**7)*exp(x/2) + (12*x**10 + 36*x**8 + 27*x**6)* 
exp(x/4))/x**10
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} \left (-24 x-16 x^3\right )} \, dx=\frac {2 \, x e^{x} - 2 \, x - e^{\left (\frac {5}{4} \, x\right )} + e^{\left (\frac {1}{4} \, x\right )}}{2 \, x^{2} - x e^{\left (\frac {1}{4} \, x\right )} + 3} \] Input:

integrate((((-4+4*x)*exp(1/4*x)^2+(-16*x^2+16*x-15)*exp(1/4*x)+16*x^3-16*x 
^2+24*x+24)*exp(x)+4*exp(1/4*x)^2+(3-16*x)*exp(1/4*x)+16*x^2-24)/(4*x^2*ex 
p(1/4*x)^2+(-16*x^3-24*x)*exp(1/4*x)+16*x^4+48*x^2+36),x, algorithm="maxim 
a")
 

Output:

(2*x*e^x - 2*x - e^(5/4*x) + e^(1/4*x))/(2*x^2 - x*e^(1/4*x) + 3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} \left (-24 x-16 x^3\right )} \, dx=\frac {2 \, x e^{x} - 2 \, x - e^{\left (\frac {5}{4} \, x\right )} + e^{\left (\frac {1}{4} \, x\right )}}{2 \, x^{2} - x e^{\left (\frac {1}{4} \, x\right )} + 3} \] Input:

integrate((((-4+4*x)*exp(1/4*x)^2+(-16*x^2+16*x-15)*exp(1/4*x)+16*x^3-16*x 
^2+24*x+24)*exp(x)+4*exp(1/4*x)^2+(3-16*x)*exp(1/4*x)+16*x^2-24)/(4*x^2*ex 
p(1/4*x)^2+(-16*x^3-24*x)*exp(1/4*x)+16*x^4+48*x^2+36),x, algorithm="giac" 
)
 

Output:

(2*x*e^x - 2*x - e^(5/4*x) + e^(1/4*x))/(2*x^2 - x*e^(1/4*x) + 3)
 

Mupad [B] (verification not implemented)

Time = 4.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 6.52 \[ \int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} \left (-24 x-16 x^3\right )} \, dx=24\,x+\frac {107\,x^4+162\,x^2+81}{x^5}+\frac {{\mathrm {e}}^x}{x}+\frac {3\,{\mathrm {e}}^{\frac {3\,x}{4}}}{x^2}+\frac {{\mathrm {e}}^{x/2}\,\left (6\,x^2+9\right )}{x^3}+\frac {{\mathrm {e}}^{x/4}\,\left (12\,x^4+36\,x^2+27\right )}{x^4}-\frac {3\,\left (32\,x^{11}-128\,x^{10}+240\,x^9-576\,x^8+718\,x^7-568\,x^6+1077\,x^5+852\,x^4+810\,x^3+1944\,x^2+243\,x+972\right )}{x^5\,\left (2\,x^2-x\,{\mathrm {e}}^{x/4}+3\right )\,\left (2\,x^3-8\,x^2+3\,x+12\right )} \] Input:

int((4*exp(x/2) + exp(x)*(24*x - exp(x/4)*(16*x^2 - 16*x + 15) + exp(x/2)* 
(4*x - 4) - 16*x^2 + 16*x^3 + 24) - exp(x/4)*(16*x - 3) + 16*x^2 - 24)/(4* 
x^2*exp(x/2) - exp(x/4)*(24*x + 16*x^3) + 48*x^2 + 16*x^4 + 36),x)
 

Output:

24*x + (162*x^2 + 107*x^4 + 81)/x^5 + exp(x)/x + (3*exp((3*x)/4))/x^2 + (e 
xp(x/2)*(6*x^2 + 9))/x^3 + (exp(x/4)*(36*x^2 + 12*x^4 + 27))/x^4 - (3*(243 
*x + 1944*x^2 + 810*x^3 + 852*x^4 + 1077*x^5 - 568*x^6 + 718*x^7 - 576*x^8 
 + 240*x^9 - 128*x^10 + 32*x^11 + 972))/(x^5*(2*x^2 - x*exp(x/4) + 3)*(3*x 
 - 8*x^2 + 2*x^3 + 12))
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} \left (-24 x-16 x^3\right )} \, dx=\frac {e^{\frac {5 x}{4}}-e^{\frac {x}{4}}-2 e^{x} x +2 x}{e^{\frac {x}{4}} x -2 x^{2}-3} \] Input:

int((((-4+4*x)*exp(1/4*x)^2+(-16*x^2+16*x-15)*exp(1/4*x)+16*x^3-16*x^2+24* 
x+24)*exp(x)+4*exp(1/4*x)^2+(3-16*x)*exp(1/4*x)+16*x^2-24)/(4*x^2*exp(1/4* 
x)^2+(-16*x^3-24*x)*exp(1/4*x)+16*x^4+48*x^2+36),x)
 

Output:

(e**((5*x)/4) - e**(x/4) - 2*e**x*x + 2*x)/(e**(x/4)*x - 2*x**2 - 3)