\(\int \frac {130 x^3+e^{2 x} (78 x+52 x^2)+e^x (208 x^2+52 x^3)+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} (72 x^7+12 x^4 \log (2))+e^x (48 x^8+24 x^5 \log (2))} \, dx\) [56]

Optimal result

Integrand size = 121, antiderivative size = 24 \[ \int \frac {130 x^3+e^{2 x} \left (78 x+52 x^2\right )+e^x \left (208 x^2+52 x^3\right )+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} \left (72 x^7+12 x^4 \log (2)\right )+e^x \left (48 x^8+24 x^5 \log (2)\right )} \, dx=5-\frac {13}{3 x^2 \left (2 x \left (e^x+x\right )^2+\log (2)\right )} \] Output:

-13/3/x^2/(ln(2)+2*x*(exp(x)+x)^2)+5
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 5.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {130 x^3+e^{2 x} \left (78 x+52 x^2\right )+e^x \left (208 x^2+52 x^3\right )+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} \left (72 x^7+12 x^4 \log (2)\right )+e^x \left (48 x^8+24 x^5 \log (2)\right )} \, dx=-\frac {13}{3 x^2 \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )} \] Input:

Integrate[(130*x^3 + E^(2*x)*(78*x + 52*x^2) + E^x*(208*x^2 + 52*x^3) + 26 
*Log[2])/(12*E^(4*x)*x^5 + 48*E^(3*x)*x^6 + 12*x^9 + 12*x^6*Log[2] + 3*x^3 
*Log[2]^2 + E^(2*x)*(72*x^7 + 12*x^4*Log[2]) + E^x*(48*x^8 + 24*x^5*Log[2] 
)),x]
 

Output:

-13/(3*x^2*(2*E^(2*x)*x + 4*E^x*x^2 + 2*x^3 + Log[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.

\(\displaystyle \int \frac {130 x^3+e^{2 x} \left (52 x^2+78 x\right )+e^x \left (52 x^3+208 x^2\right )+26 \log (2)}{12 x^9+48 e^{3 x} x^6+12 x^6 \log (2)+12 e^{4 x} x^5+3 x^3 \log ^2(2)+e^x \left (48 x^8+24 x^5 \log (2)\right )+e^{2 x} \left (72 x^7+12 x^4 \log (2)\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {26 \left (5 x^3+2 e^x (x+4) x^2+e^{2 x} (2 x+3) x+\log (2)\right )}{3 x^3 \left (2 x^3+4 e^x x^2+2 e^{2 x} x+\log (2)\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {26}{3} \int \frac {5 x^3+2 e^x (x+4) x^2+e^{2 x} (2 x+3) x+\log (2)}{x^3 \left (2 x^3+4 e^x x^2+2 e^{2 x} x+\log (2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {26}{3} \int \left (\frac {2 x+3}{x^3 \left (4 x^3+8 e^x x^2+4 e^{2 x} x+\log (4)\right )}-\frac {4 x^4+4 e^x x^3-4 x^3-4 e^x x^2+\log (4) x+\log (2)}{2 x^3 \left (2 x^3+4 e^x x^2+2 e^{2 x} x+\log (2)\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {26}{3} \left (2 \int \frac {1}{\left (2 x^3+4 e^x x^2+2 e^{2 x} x+\log (2)\right )^2}dx-2 \int \frac {e^x}{\left (2 x^3+4 e^x x^2+2 e^{2 x} x+\log (2)\right )^2}dx-\frac {1}{2} \log (2) \int \frac {1}{x^3 \left (2 x^3+4 e^x x^2+2 e^{2 x} x+\log (2)\right )^2}dx-\frac {1}{2} \log (4) \int \frac {1}{x^2 \left (2 x^3+4 e^x x^2+2 e^{2 x} x+\log (2)\right )^2}dx+2 \int \frac {e^x}{x \left (2 x^3+4 e^x x^2+2 e^{2 x} x+\log (2)\right )^2}dx-2 \int \frac {x}{\left (2 x^3+4 e^x x^2+2 e^{2 x} x+\log (2)\right )^2}dx+3 \int \frac {1}{x^3 \left (4 x^3+8 e^x x^2+4 e^{2 x} x+\log (4)\right )}dx+2 \int \frac {1}{x^2 \left (4 x^3+8 e^x x^2+4 e^{2 x} x+\log (4)\right )}dx\right )\)

Input:

Int[(130*x^3 + E^(2*x)*(78*x + 52*x^2) + E^x*(208*x^2 + 52*x^3) + 26*Log[2 
])/(12*E^(4*x)*x^5 + 48*E^(3*x)*x^6 + 12*x^9 + 12*x^6*Log[2] + 3*x^3*Log[2 
]^2 + E^(2*x)*(72*x^7 + 12*x^4*Log[2]) + E^x*(48*x^8 + 24*x^5*Log[2])),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25

method	result	size
risch	\(-\frac {13}{3 x^{2} \left (2 x \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} x^{2}+2 x^{3}+\ln \left (2\right )\right )}\)	\(30\)
parallelrisch	\(-\frac {13}{3 x^{2} \left (2 x \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} x^{2}+2 x^{3}+\ln \left (2\right )\right )}\)	\(30\)

Input:

int(((52*x^2+78*x)*exp(x)^2+(52*x^3+208*x^2)*exp(x)+26*ln(2)+130*x^3)/(12* 
x^5*exp(x)^4+48*x^6*exp(x)^3+(12*x^4*ln(2)+72*x^7)*exp(x)^2+(24*x^5*ln(2)+ 
48*x^8)*exp(x)+3*x^3*ln(2)^2+12*x^6*ln(2)+12*x^9),x,method=_RETURNVERBOSE)
 

Output:

-13/3/x^2/(2*x*exp(x)^2+4*exp(x)*x^2+2*x^3+ln(2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {130 x^3+e^{2 x} \left (78 x+52 x^2\right )+e^x \left (208 x^2+52 x^3\right )+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} \left (72 x^7+12 x^4 \log (2)\right )+e^x \left (48 x^8+24 x^5 \log (2)\right )} \, dx=-\frac {13}{3 \, {\left (2 \, x^{5} + 4 \, x^{4} e^{x} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{2} \log \left (2\right )\right )}} \] Input:

integrate(((52*x^2+78*x)*exp(x)^2+(52*x^3+208*x^2)*exp(x)+26*log(2)+130*x^ 
3)/(12*x^5*exp(x)^4+48*x^6*exp(x)^3+(12*x^4*log(2)+72*x^7)*exp(x)^2+(24*x^ 
5*log(2)+48*x^8)*exp(x)+3*x^3*log(2)^2+12*x^6*log(2)+12*x^9),x, algorithm= 
"fricas")
 

Output:

-13/3/(2*x^5 + 4*x^4*e^x + 2*x^3*e^(2*x) + x^2*log(2))
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {130 x^3+e^{2 x} \left (78 x+52 x^2\right )+e^x \left (208 x^2+52 x^3\right )+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} \left (72 x^7+12 x^4 \log (2)\right )+e^x \left (48 x^8+24 x^5 \log (2)\right )} \, dx=- \frac {13}{6 x^{5} + 12 x^{4} e^{x} + 6 x^{3} e^{2 x} + 3 x^{2} \log {\left (2 \right )}} \] Input:

integrate(((52*x**2+78*x)*exp(x)**2+(52*x**3+208*x**2)*exp(x)+26*ln(2)+130 
*x**3)/(12*x**5*exp(x)**4+48*x**6*exp(x)**3+(12*x**4*ln(2)+72*x**7)*exp(x) 
**2+(24*x**5*ln(2)+48*x**8)*exp(x)+3*x**3*ln(2)**2+12*x**6*ln(2)+12*x**9), 
x)
 

Output:

-13/(6*x**5 + 12*x**4*exp(x) + 6*x**3*exp(2*x) + 3*x**2*log(2))
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {130 x^3+e^{2 x} \left (78 x+52 x^2\right )+e^x \left (208 x^2+52 x^3\right )+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} \left (72 x^7+12 x^4 \log (2)\right )+e^x \left (48 x^8+24 x^5 \log (2)\right )} \, dx=-\frac {13}{3 \, {\left (2 \, x^{5} + 4 \, x^{4} e^{x} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{2} \log \left (2\right )\right )}} \] Input:

integrate(((52*x^2+78*x)*exp(x)^2+(52*x^3+208*x^2)*exp(x)+26*log(2)+130*x^ 
3)/(12*x^5*exp(x)^4+48*x^6*exp(x)^3+(12*x^4*log(2)+72*x^7)*exp(x)^2+(24*x^ 
5*log(2)+48*x^8)*exp(x)+3*x^3*log(2)^2+12*x^6*log(2)+12*x^9),x, algorithm= 
"maxima")
 

Output:

-13/3/(2*x^5 + 4*x^4*e^x + 2*x^3*e^(2*x) + x^2*log(2))
 

Giac [A] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {130 x^3+e^{2 x} \left (78 x+52 x^2\right )+e^x \left (208 x^2+52 x^3\right )+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} \left (72 x^7+12 x^4 \log (2)\right )+e^x \left (48 x^8+24 x^5 \log (2)\right )} \, dx=-\frac {26}{3 \, {\left (2 \, x^{5} + 4 \, x^{4} e^{x} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{2} \log \left (2\right )\right )}} \] Input:

integrate(((52*x^2+78*x)*exp(x)^2+(52*x^3+208*x^2)*exp(x)+26*log(2)+130*x^ 
3)/(12*x^5*exp(x)^4+48*x^6*exp(x)^3+(12*x^4*log(2)+72*x^7)*exp(x)^2+(24*x^ 
5*log(2)+48*x^8)*exp(x)+3*x^3*log(2)^2+12*x^6*log(2)+12*x^9),x, algorithm= 
"giac")
 

Output:

-26/3/(2*x^5 + 4*x^4*e^x + 2*x^3*e^(2*x) + x^2*log(2))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {130 x^3+e^{2 x} \left (78 x+52 x^2\right )+e^x \left (208 x^2+52 x^3\right )+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} \left (72 x^7+12 x^4 \log (2)\right )+e^x \left (48 x^8+24 x^5 \log (2)\right )} \, dx=\int \frac {26\,\ln \left (2\right )+{\mathrm {e}}^{2\,x}\,\left (52\,x^2+78\,x\right )+{\mathrm {e}}^x\,\left (52\,x^3+208\,x^2\right )+130\,x^3}{3\,x^3\,{\ln \left (2\right )}^2+12\,x^5\,{\mathrm {e}}^{4\,x}+48\,x^6\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^x\,\left (48\,x^8+24\,\ln \left (2\right )\,x^5\right )+12\,x^6\,\ln \left (2\right )+{\mathrm {e}}^{2\,x}\,\left (72\,x^7+12\,\ln \left (2\right )\,x^4\right )+12\,x^9} \,d x \] Input:

int((26*log(2) + exp(2*x)*(78*x + 52*x^2) + exp(x)*(208*x^2 + 52*x^3) + 13 
0*x^3)/(3*x^3*log(2)^2 + 12*x^5*exp(4*x) + 48*x^6*exp(3*x) + exp(x)*(24*x^ 
5*log(2) + 48*x^8) + 12*x^6*log(2) + exp(2*x)*(12*x^4*log(2) + 72*x^7) + 1 
2*x^9),x)
 

Output:

int((26*log(2) + exp(2*x)*(78*x + 52*x^2) + exp(x)*(208*x^2 + 52*x^3) + 13 
0*x^3)/(3*x^3*log(2)^2 + 12*x^5*exp(4*x) + 48*x^6*exp(3*x) + exp(x)*(24*x^ 
5*log(2) + 48*x^8) + 12*x^6*log(2) + exp(2*x)*(12*x^4*log(2) + 72*x^7) + 1 
2*x^9), x)
 

Reduce [F]

\[ \int \frac {130 x^3+e^{2 x} \left (78 x+52 x^2\right )+e^x \left (208 x^2+52 x^3\right )+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} \left (72 x^7+12 x^4 \log (2)\right )+e^x \left (48 x^8+24 x^5 \log (2)\right )} \, dx=26 \left (\int \frac {e^{2 x}}{4 e^{4 x} x^{4}+16 e^{3 x} x^{5}+4 e^{2 x} \mathrm {log}\left (2\right ) x^{3}+24 e^{2 x} x^{6}+8 e^{x} \mathrm {log}\left (2\right ) x^{4}+16 e^{x} x^{7}+\mathrm {log}\left (2\right )^{2} x^{2}+4 \,\mathrm {log}\left (2\right ) x^{5}+4 x^{8}}d x \right )+\frac {52 \left (\int \frac {e^{2 x}}{4 e^{4 x} x^{3}+16 e^{3 x} x^{4}+4 e^{2 x} \mathrm {log}\left (2\right ) x^{2}+24 e^{2 x} x^{5}+8 e^{x} \mathrm {log}\left (2\right ) x^{3}+16 e^{x} x^{6}+\mathrm {log}\left (2\right )^{2} x +4 \,\mathrm {log}\left (2\right ) x^{4}+4 x^{7}}d x \right )}{3}+\frac {208 \left (\int \frac {e^{x}}{4 e^{4 x} x^{3}+16 e^{3 x} x^{4}+4 e^{2 x} \mathrm {log}\left (2\right ) x^{2}+24 e^{2 x} x^{5}+8 e^{x} \mathrm {log}\left (2\right ) x^{3}+16 e^{x} x^{6}+\mathrm {log}\left (2\right )^{2} x +4 \,\mathrm {log}\left (2\right ) x^{4}+4 x^{7}}d x \right )}{3}+\frac {52 \left (\int \frac {e^{x}}{4 e^{4 x} x^{2}+16 e^{3 x} x^{3}+4 e^{2 x} \mathrm {log}\left (2\right ) x +24 e^{2 x} x^{4}+8 e^{x} \mathrm {log}\left (2\right ) x^{2}+16 e^{x} x^{5}+\mathrm {log}\left (2\right )^{2}+4 \,\mathrm {log}\left (2\right ) x^{3}+4 x^{6}}d x \right )}{3}+\frac {26 \left (\int \frac {1}{4 e^{4 x} x^{5}+16 e^{3 x} x^{6}+4 e^{2 x} \mathrm {log}\left (2\right ) x^{4}+24 e^{2 x} x^{7}+8 e^{x} \mathrm {log}\left (2\right ) x^{5}+16 e^{x} x^{8}+\mathrm {log}\left (2\right )^{2} x^{3}+4 \,\mathrm {log}\left (2\right ) x^{6}+4 x^{9}}d x \right ) \mathrm {log}\left (2\right )}{3}+\frac {130 \left (\int \frac {1}{4 e^{4 x} x^{2}+16 e^{3 x} x^{3}+4 e^{2 x} \mathrm {log}\left (2\right ) x +24 e^{2 x} x^{4}+8 e^{x} \mathrm {log}\left (2\right ) x^{2}+16 e^{x} x^{5}+\mathrm {log}\left (2\right )^{2}+4 \,\mathrm {log}\left (2\right ) x^{3}+4 x^{6}}d x \right )}{3} \] Input:

int(((52*x^2+78*x)*exp(x)^2+(52*x^3+208*x^2)*exp(x)+26*log(2)+130*x^3)/(12 
*x^5*exp(x)^4+48*x^6*exp(x)^3+(12*x^4*log(2)+72*x^7)*exp(x)^2+(24*x^5*log( 
2)+48*x^8)*exp(x)+3*x^3*log(2)^2+12*x^6*log(2)+12*x^9),x)
 

Output:

(26*(3*int(e**(2*x)/(4*e**(4*x)*x**4 + 16*e**(3*x)*x**5 + 4*e**(2*x)*log(2 
)*x**3 + 24*e**(2*x)*x**6 + 8*e**x*log(2)*x**4 + 16*e**x*x**7 + log(2)**2* 
x**2 + 4*log(2)*x**5 + 4*x**8),x) + 2*int(e**(2*x)/(4*e**(4*x)*x**3 + 16*e 
**(3*x)*x**4 + 4*e**(2*x)*log(2)*x**2 + 24*e**(2*x)*x**5 + 8*e**x*log(2)*x 
**3 + 16*e**x*x**6 + log(2)**2*x + 4*log(2)*x**4 + 4*x**7),x) + 8*int(e**x 
/(4*e**(4*x)*x**3 + 16*e**(3*x)*x**4 + 4*e**(2*x)*log(2)*x**2 + 24*e**(2*x 
)*x**5 + 8*e**x*log(2)*x**3 + 16*e**x*x**6 + log(2)**2*x + 4*log(2)*x**4 + 
 4*x**7),x) + 2*int(e**x/(4*e**(4*x)*x**2 + 16*e**(3*x)*x**3 + 4*e**(2*x)* 
log(2)*x + 24*e**(2*x)*x**4 + 8*e**x*log(2)*x**2 + 16*e**x*x**5 + log(2)** 
2 + 4*log(2)*x**3 + 4*x**6),x) + int(1/(4*e**(4*x)*x**5 + 16*e**(3*x)*x**6 
 + 4*e**(2*x)*log(2)*x**4 + 24*e**(2*x)*x**7 + 8*e**x*log(2)*x**5 + 16*e** 
x*x**8 + log(2)**2*x**3 + 4*log(2)*x**6 + 4*x**9),x)*log(2) + 5*int(1/(4*e 
**(4*x)*x**2 + 16*e**(3*x)*x**3 + 4*e**(2*x)*log(2)*x + 24*e**(2*x)*x**4 + 
 8*e**x*log(2)*x**2 + 16*e**x*x**5 + log(2)**2 + 4*log(2)*x**3 + 4*x**6),x 
)))/3