\(\int \frac {e^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 (1280 x-160 x^2+5 x^3)}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} (768 x^3-96 x^4+3 x^5)+e^5 (-1344 x^4+180 x^5-6 x^6)+e^{3/x} (84 x^4-6 x^5+e^5 (-96 x^3+6 x^4))} \, dx\) [1073]

Optimal result

Integrand size = 152, antiderivative size = 35 \[ \int \frac {e^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 \left (1280 x-160 x^2+5 x^3\right )}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} \left (768 x^3-96 x^4+3 x^5\right )+e^5 \left (-1344 x^4+180 x^5-6 x^6\right )+e^{3/x} \left (84 x^4-6 x^5+e^5 \left (-96 x^3+6 x^4\right )\right )} \, dx=\frac {5}{3 x \left (-e^5+\frac {e^{3/x}-2 x}{16-x}+x\right )} \] Output:

5/3/(x+(exp(3/x)-2*x)/(16-x)-exp(5))/x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {e^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 \left (1280 x-160 x^2+5 x^3\right )}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} \left (768 x^3-96 x^4+3 x^5\right )+e^5 \left (-1344 x^4+180 x^5-6 x^6\right )+e^{3/x} \left (84 x^4-6 x^5+e^5 \left (-96 x^3+6 x^4\right )\right )} \, dx=-\frac {5 (-16+x)}{3 x \left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )} \] Input:

Integrate[(E^(3/x)*(240 - 95*x) - 2240*x^2 + 310*x^3 - 10*x^4 + E^5*(1280* 
x - 160*x^2 + 5*x^3))/(3*E^(6/x)*x^3 + 588*x^5 - 84*x^6 + 3*x^7 + E^10*(76 
8*x^3 - 96*x^4 + 3*x^5) + E^5*(-1344*x^4 + 180*x^5 - 6*x^6) + E^(3/x)*(84* 
x^4 - 6*x^5 + E^5*(-96*x^3 + 6*x^4))),x]
 

Output:

(-5*(-16 + x))/(3*x*(E^(3/x) + E^5*(-16 + x) - (-14 + x)*x))
 

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.

\(\displaystyle \int \frac {-10 x^4+310 x^3-2240 x^2+e^5 \left (5 x^3-160 x^2+1280 x\right )+e^{3/x} (240-95 x)}{3 x^7-84 x^6+588 x^5+3 e^{6/x} x^3+e^5 \left (-6 x^6+180 x^5-1344 x^4\right )+e^{10} \left (3 x^5-96 x^4+768 x^3\right )+e^{3/x} \left (-6 x^5+84 x^4+e^5 \left (6 x^4-96 x^3\right )\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {5 \left (-2 x^2 \left (x^2-31 x+224\right )+e^5 x (x-16)^2-e^{3/x} (19 x-48)\right )}{3 x^3 \left (e^5 (x-16)+e^{3/x}-(x-14) x\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{3} \int \frac {e^5 x (16-x)^2+e^{3/x} (48-19 x)-2 x^2 \left (x^2-31 x+224\right )}{x^3 \left (-e^5 (16-x)+e^{3/x}+(14-x) x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {5}{3} \int \left (\frac {19 x-48}{x^3 \left (x^2-14 \left (1+\frac {e^5}{14}\right ) x-e^{3/x}+16 e^5\right )}+\frac {(16-x) \left (2 x^3-\left (11+e^5\right ) x^2-3 \left (14+e^5\right ) x+48 e^5\right )}{x^3 \left (x^2-14 \left (1+\frac {e^5}{14}\right ) x-e^{3/x}+16 e^5\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{3} \left (\left (43+e^5\right ) \int \frac {1}{\left (x^2-14 \left (1+\frac {e^5}{14}\right ) x-e^{3/x}+16 e^5\right )^2}dx-96 \left (7+e^5\right ) \int \frac {1}{x^2 \left (x^2-14 \left (1+\frac {e^5}{14}\right ) x-e^{3/x}+16 e^5\right )^2}dx-\left (134+13 e^5\right ) \int \frac {1}{x \left (x^2-14 \left (1+\frac {e^5}{14}\right ) x-e^{3/x}+16 e^5\right )^2}dx-2 \int \frac {x}{\left (x^2-14 \left (1+\frac {e^5}{14}\right ) x-e^{3/x}+16 e^5\right )^2}dx+19 \int \frac {1}{x^2 \left (x^2-14 \left (1+\frac {e^5}{14}\right ) x-e^{3/x}+16 e^5\right )}dx+48 \int \frac {1}{x^3 \left (-x^2+14 \left (1+\frac {e^5}{14}\right ) x+e^{3/x}-16 e^5\right )}dx+768 e^5 \int \frac {1}{x^3 \left (x^2-14 \left (1+\frac {e^5}{14}\right ) x-e^{3/x}+16 e^5\right )^2}dx\right )\)

Input:

Int[(E^(3/x)*(240 - 95*x) - 2240*x^2 + 310*x^3 - 10*x^4 + E^5*(1280*x - 16 
0*x^2 + 5*x^3))/(3*E^(6/x)*x^3 + 588*x^5 - 84*x^6 + 3*x^7 + E^10*(768*x^3 
- 96*x^4 + 3*x^5) + E^5*(-1344*x^4 + 180*x^5 - 6*x^6) + E^(3/x)*(84*x^4 - 
6*x^5 + E^5*(-96*x^3 + 6*x^4))),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97

method	result	size
risch	\(-\frac {5 \left (x -16\right )}{3 x \left (x \,{\mathrm e}^{5}-x^{2}-16 \,{\mathrm e}^{5}+{\mathrm e}^{\frac {3}{x}}+14 x \right )}\)	\(34\)
parallelrisch	\(\frac {80-5 x}{3 x \left (x \,{\mathrm e}^{5}-x^{2}-16 \,{\mathrm e}^{5}+{\mathrm e}^{\frac {3}{x}}+14 x \right )}\)	\(36\)
norman	\(\frac {-\frac {5}{3} x^{2}+\frac {80}{3} x}{x^{2} \left (x \,{\mathrm e}^{5}-x^{2}-16 \,{\mathrm e}^{5}+{\mathrm e}^{\frac {3}{x}}+14 x \right )}\)	\(39\)

Input:

int(((-95*x+240)*exp(3/x)+(5*x^3-160*x^2+1280*x)*exp(5)-10*x^4+310*x^3-224 
0*x^2)/(3*x^3*exp(3/x)^2+((6*x^4-96*x^3)*exp(5)-6*x^5+84*x^4)*exp(3/x)+(3* 
x^5-96*x^4+768*x^3)*exp(5)^2+(-6*x^6+180*x^5-1344*x^4)*exp(5)+3*x^7-84*x^6 
+588*x^5),x,method=_RETURNVERBOSE)
 

Output:

-5/3*(x-16)/x/(x*exp(5)-x^2-16*exp(5)+exp(3/x)+14*x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \frac {e^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 \left (1280 x-160 x^2+5 x^3\right )}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} \left (768 x^3-96 x^4+3 x^5\right )+e^5 \left (-1344 x^4+180 x^5-6 x^6\right )+e^{3/x} \left (84 x^4-6 x^5+e^5 \left (-96 x^3+6 x^4\right )\right )} \, dx=\frac {5 \, {\left (x - 16\right )}}{3 \, {\left (x^{3} - 14 \, x^{2} - {\left (x^{2} - 16 \, x\right )} e^{5} - x e^{\frac {3}{x}}\right )}} \] Input:

integrate(((-95*x+240)*exp(3/x)+(5*x^3-160*x^2+1280*x)*exp(5)-10*x^4+310*x 
^3-2240*x^2)/(3*x^3*exp(3/x)^2+((6*x^4-96*x^3)*exp(5)-6*x^5+84*x^4)*exp(3/ 
x)+(3*x^5-96*x^4+768*x^3)*exp(5)^2+(-6*x^6+180*x^5-1344*x^4)*exp(5)+3*x^7- 
84*x^6+588*x^5),x, algorithm="fricas")
 

Output:

5/3*(x - 16)/(x^3 - 14*x^2 - (x^2 - 16*x)*e^5 - x*e^(3/x))
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {e^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 \left (1280 x-160 x^2+5 x^3\right )}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} \left (768 x^3-96 x^4+3 x^5\right )+e^5 \left (-1344 x^4+180 x^5-6 x^6\right )+e^{3/x} \left (84 x^4-6 x^5+e^5 \left (-96 x^3+6 x^4\right )\right )} \, dx=\frac {80 - 5 x}{- 3 x^{3} + 42 x^{2} + 3 x^{2} e^{5} + 3 x e^{\frac {3}{x}} - 48 x e^{5}} \] Input:

integrate(((-95*x+240)*exp(3/x)+(5*x**3-160*x**2+1280*x)*exp(5)-10*x**4+31 
0*x**3-2240*x**2)/(3*x**3*exp(3/x)**2+((6*x**4-96*x**3)*exp(5)-6*x**5+84*x 
**4)*exp(3/x)+(3*x**5-96*x**4+768*x**3)*exp(5)**2+(-6*x**6+180*x**5-1344*x 
**4)*exp(5)+3*x**7-84*x**6+588*x**5),x)
 

Output:

(80 - 5*x)/(-3*x**3 + 42*x**2 + 3*x**2*exp(5) + 3*x*exp(3/x) - 48*x*exp(5) 
)
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {e^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 \left (1280 x-160 x^2+5 x^3\right )}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} \left (768 x^3-96 x^4+3 x^5\right )+e^5 \left (-1344 x^4+180 x^5-6 x^6\right )+e^{3/x} \left (84 x^4-6 x^5+e^5 \left (-96 x^3+6 x^4\right )\right )} \, dx=\frac {5 \, {\left (x - 16\right )}}{3 \, {\left (x^{3} - x^{2} {\left (e^{5} + 14\right )} + 16 \, x e^{5} - x e^{\frac {3}{x}}\right )}} \] Input:

integrate(((-95*x+240)*exp(3/x)+(5*x^3-160*x^2+1280*x)*exp(5)-10*x^4+310*x 
^3-2240*x^2)/(3*x^3*exp(3/x)^2+((6*x^4-96*x^3)*exp(5)-6*x^5+84*x^4)*exp(3/ 
x)+(3*x^5-96*x^4+768*x^3)*exp(5)^2+(-6*x^6+180*x^5-1344*x^4)*exp(5)+3*x^7- 
84*x^6+588*x^5),x, algorithm="maxima")
 

Output:

5/3*(x - 16)/(x^3 - x^2*(e^5 + 14) + 16*x*e^5 - x*e^(3/x))
 

Giac [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {e^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 \left (1280 x-160 x^2+5 x^3\right )}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} \left (768 x^3-96 x^4+3 x^5\right )+e^5 \left (-1344 x^4+180 x^5-6 x^6\right )+e^{3/x} \left (84 x^4-6 x^5+e^5 \left (-96 x^3+6 x^4\right )\right )} \, dx=-\frac {5 \, {\left (\frac {1}{x^{2}} - \frac {16}{x^{3}}\right )}}{3 \, {\left (\frac {e^{5}}{x} + \frac {14}{x} - \frac {16 \, e^{5}}{x^{2}} + \frac {e^{\frac {3}{x}}}{x^{2}} - 1\right )}} \] Input:

integrate(((-95*x+240)*exp(3/x)+(5*x^3-160*x^2+1280*x)*exp(5)-10*x^4+310*x 
^3-2240*x^2)/(3*x^3*exp(3/x)^2+((6*x^4-96*x^3)*exp(5)-6*x^5+84*x^4)*exp(3/ 
x)+(3*x^5-96*x^4+768*x^3)*exp(5)^2+(-6*x^6+180*x^5-1344*x^4)*exp(5)+3*x^7- 
84*x^6+588*x^5),x, algorithm="giac")
 

Output:

-5/3*(1/x^2 - 16/x^3)/(e^5/x + 14/x - 16*e^5/x^2 + e^(3/x)/x^2 - 1)
 

Mupad [B] (verification not implemented)

Time = 3.49 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.03 \[ \int \frac {e^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 \left (1280 x-160 x^2+5 x^3\right )}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} \left (768 x^3-96 x^4+3 x^5\right )+e^5 \left (-1344 x^4+180 x^5-6 x^6\right )+e^{3/x} \left (84 x^4-6 x^5+e^5 \left (-96 x^3+6 x^4\right )\right )} \, dx=\frac {\frac {10\,x^5}{3}+\left (-\frac {5\,{\mathrm {e}}^5}{3}-\frac {215}{3}\right )\,x^4+\left (\frac {65\,{\mathrm {e}}^5}{3}+\frac {670}{3}\right )\,x^3+\left (160\,{\mathrm {e}}^5+1120\right )\,x^2-1280\,{\mathrm {e}}^5\,x}{\left (14\,x-16\,{\mathrm {e}}^5+{\mathrm {e}}^{3/x}+x\,{\mathrm {e}}^5-x^2\right )\,\left (3\,x^3\,{\mathrm {e}}^5-48\,x^2\,{\mathrm {e}}^5+x^4\,{\mathrm {e}}^5+42\,x^3+11\,x^4-2\,x^5\right )} \] Input:

int(-(exp(3/x)*(95*x - 240) - exp(5)*(1280*x - 160*x^2 + 5*x^3) + 2240*x^2 
 - 310*x^3 + 10*x^4)/(exp(10)*(768*x^3 - 96*x^4 + 3*x^5) - exp(3/x)*(exp(5 
)*(96*x^3 - 6*x^4) - 84*x^4 + 6*x^5) - exp(5)*(1344*x^4 - 180*x^5 + 6*x^6) 
 + 3*x^3*exp(6/x) + 588*x^5 - 84*x^6 + 3*x^7),x)
 

Output:

(x^3*((65*exp(5))/3 + 670/3) - x^4*((5*exp(5))/3 + 215/3) - 1280*x*exp(5) 
+ x^2*(160*exp(5) + 1120) + (10*x^5)/3)/((14*x - 16*exp(5) + exp(3/x) + x* 
exp(5) - x^2)*(3*x^3*exp(5) - 48*x^2*exp(5) + x^4*exp(5) + 42*x^3 + 11*x^4 
 - 2*x^5))
 

Reduce [F]

\[ \int \frac {e^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 \left (1280 x-160 x^2+5 x^3\right )}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} \left (768 x^3-96 x^4+3 x^5\right )+e^5 \left (-1344 x^4+180 x^5-6 x^6\right )+e^{3/x} \left (84 x^4-6 x^5+e^5 \left (-96 x^3+6 x^4\right )\right )} \, dx =\text {Too large to display} \] Input:

int(((-95*x+240)*exp(3/x)+(5*x^3-160*x^2+1280*x)*exp(5)-10*x^4+310*x^3-224 
0*x^2)/(3*x^3*exp(3/x)^2+((6*x^4-96*x^3)*exp(5)-6*x^5+84*x^4)*exp(3/x)+(3* 
x^5-96*x^4+768*x^3)*exp(5)^2+(-6*x^6+180*x^5-1344*x^4)*exp(5)+3*x^7-84*x^6 
+588*x^5),x)
 

Output:

(5*(48*int(e**(3/x)/(e**(6/x)*x**3 + 2*e**(3/x)*e**5*x**4 - 32*e**(3/x)*e* 
*5*x**3 - 2*e**(3/x)*x**5 + 28*e**(3/x)*x**4 + e**10*x**5 - 32*e**10*x**4 
+ 256*e**10*x**3 - 2*e**5*x**6 + 60*e**5*x**5 - 448*e**5*x**4 + x**7 - 28* 
x**6 + 196*x**5),x) - 19*int(e**(3/x)/(e**(6/x)*x**2 + 2*e**(3/x)*e**5*x** 
3 - 32*e**(3/x)*e**5*x**2 - 2*e**(3/x)*x**4 + 28*e**(3/x)*x**3 + e**10*x** 
4 - 32*e**10*x**3 + 256*e**10*x**2 - 2*e**5*x**5 + 60*e**5*x**4 - 448*e**5 
*x**3 + x**6 - 28*x**5 + 196*x**4),x) - 2*int(x/(e**(6/x) + 2*e**(3/x)*e** 
5*x - 32*e**(3/x)*e**5 - 2*e**(3/x)*x**2 + 28*e**(3/x)*x + e**10*x**2 - 32 
*e**10*x + 256*e**10 - 2*e**5*x**3 + 60*e**5*x**2 - 448*e**5*x + x**4 - 28 
*x**3 + 196*x**2),x) + int(1/(e**(6/x) + 2*e**(3/x)*e**5*x - 32*e**(3/x)*e 
**5 - 2*e**(3/x)*x**2 + 28*e**(3/x)*x + e**10*x**2 - 32*e**10*x + 256*e**1 
0 - 2*e**5*x**3 + 60*e**5*x**2 - 448*e**5*x + x**4 - 28*x**3 + 196*x**2),x 
)*e**5 + 62*int(1/(e**(6/x) + 2*e**(3/x)*e**5*x - 32*e**(3/x)*e**5 - 2*e** 
(3/x)*x**2 + 28*e**(3/x)*x + e**10*x**2 - 32*e**10*x + 256*e**10 - 2*e**5* 
x**3 + 60*e**5*x**2 - 448*e**5*x + x**4 - 28*x**3 + 196*x**2),x) + 256*int 
(1/(e**(6/x)*x**2 + 2*e**(3/x)*e**5*x**3 - 32*e**(3/x)*e**5*x**2 - 2*e**(3 
/x)*x**4 + 28*e**(3/x)*x**3 + e**10*x**4 - 32*e**10*x**3 + 256*e**10*x**2 
- 2*e**5*x**5 + 60*e**5*x**4 - 448*e**5*x**3 + x**6 - 28*x**5 + 196*x**4), 
x)*e**5 - 32*int(1/(e**(6/x)*x + 2*e**(3/x)*e**5*x**2 - 32*e**(3/x)*e**5*x 
 - 2*e**(3/x)*x**3 + 28*e**(3/x)*x**2 + e**10*x**3 - 32*e**10*x**2 + 25...