\(\int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x (-720+670 x+25 x^2-960 x^3+240 x^4) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx\) [2408]

Optimal result

Integrand size = 91, antiderivative size = 24 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=x+\frac {e^x \log (3)}{x \left (x+\frac {48}{5} \left (3+x^3\right )\right )} \] Output:

ln(3)*exp(x)/x/(x+48/5*x^3+144/5)+x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=x+\frac {5 e^x \log (3)}{x \left (144+5 x+48 x^3\right )} \] Input:

Integrate[(20736*x^2 + 1440*x^3 + 25*x^4 + 13824*x^5 + 480*x^6 + 2304*x^8 
+ E^x*(-720 + 670*x + 25*x^2 - 960*x^3 + 240*x^4)*Log[3])/(20736*x^2 + 144 
0*x^3 + 25*x^4 + 13824*x^5 + 480*x^6 + 2304*x^8),x]
 

Output:

x + (5*E^x*Log[3])/(x*(144 + 5*x + 48*x^3))
 

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.

\(\displaystyle \int \frac {2304 x^8+480 x^6+13824 x^5+25 x^4+1440 x^3+20736 x^2+e^x \left (240 x^4-960 x^3+25 x^2+670 x-720\right ) \log (3)}{2304 x^8+480 x^6+13824 x^5+25 x^4+1440 x^3+20736 x^2} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {2304 x^8+480 x^6+13824 x^5+25 x^4+1440 x^3+20736 x^2+e^x \left (240 x^4-960 x^3+25 x^2+670 x-720\right ) \log (3)}{x^2 \left (2304 x^6+480 x^4+13824 x^3+25 x^2+1440 x+20736\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \frac {2304 x^8+480 x^6+13824 x^5+25 x^4+1440 x^3+20736 x^2+e^x \left (240 x^4-960 x^3+25 x^2+670 x-720\right ) \log (3)}{x^2 \left (48 x^3+5 x+144\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {13824 x^3}{\left (48 x^3+5 x+144\right )^2}+\frac {1440 x}{\left (48 x^3+5 x+144\right )^2}+\frac {20736}{\left (48 x^3+5 x+144\right )^2}+\frac {2304 x^6}{\left (48 x^3+5 x+144\right )^2}+\frac {480 x^4}{\left (48 x^3+5 x+144\right )^2}+\frac {25 x^2}{\left (48 x^3+5 x+144\right )^2}+\frac {5 e^x \left (48 x^4-192 x^3+5 x^2+134 x-144\right ) \log (3)}{\left (48 x^3+5 x+144\right )^2 x^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {13824 x^3}{\left (48 x^3+5 x+144\right )^2}+\frac {1440 x}{\left (48 x^3+5 x+144\right )^2}+\frac {20736}{\left (48 x^3+5 x+144\right )^2}+\frac {2304 x^6}{\left (48 x^3+5 x+144\right )^2}+\frac {480 x^4}{\left (48 x^3+5 x+144\right )^2}+\frac {25 x^2}{\left (48 x^3+5 x+144\right )^2}+\frac {5 e^x \left (48 x^4-192 x^3+5 x^2+134 x-144\right ) \log (3)}{\left (48 x^3+5 x+144\right )^2 x^2}\right )dx\)

Input:

Int[(20736*x^2 + 1440*x^3 + 25*x^4 + 13824*x^5 + 480*x^6 + 2304*x^8 + E^x* 
(-720 + 670*x + 25*x^2 - 960*x^3 + 240*x^4)*Log[3])/(20736*x^2 + 1440*x^3 
+ 25*x^4 + 13824*x^5 + 480*x^6 + 2304*x^8),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

method	result	size
risch	\(x +\frac {5 \ln \left (3\right ) {\mathrm e}^{x}}{x \left (48 x^{3}+5 x +144\right )}\)	\(24\)
parallelrisch	\(\frac {2304 x^{5}+240 x^{3}+240 \ln \left (3\right ) {\mathrm e}^{x}+6912 x^{2}}{48 x \left (48 x^{3}+5 x +144\right )}\)	\(40\)
norman	\(\frac {-\frac {20736 x}{5}-\frac {6912 x^{4}}{5}+5 x^{3}+48 x^{5}+5 \ln \left (3\right ) {\mathrm e}^{x}}{x \left (48 x^{3}+5 x +144\right )}\)	\(42\)
parts	\(x -\frac {25 \ln \left (3\right ) \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (48 \textit {\_Z}^{3}+5 \textit {\_Z} +144\right )}{\sum }\frac {\left (360 \textit {\_R1}^{2}-15912 \textit {\_R1} +31129\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-x +\textit {\_R1} \right )}{144 \textit {\_R1}^{2}+5}\right )}{6718589}+\frac {5 \ln \left (3\right ) {\mathrm e}^{x}}{x \left (48 x^{3}+5 x +144\right )}+\frac {5 \ln \left (3\right ) \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (48 \textit {\_Z}^{3}+5 \textit {\_Z} +144\right )}{\sum }\frac {\left (9352563888 \textit {\_R1}^{2}-30971987712 \textit {\_R1} [phone]\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-x +\textit {\_R1} \right )}{144 \textit {\_R1}^{2}+5}\right )}{69658330752}+\frac {335 \ln \left (3\right ) \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (48 \textit {\_Z}^{3}+5 \textit {\_Z} +144\right )}{\sum }\frac {\left (322665072 \textit {\_R1}^{2}-7637760 \textit {\_R1} [phone]\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-x +\textit {\_R1} \right )}{144 \textit {\_R1}^{2}+5}\right )}{69658330752}-\frac {480 \ln \left (3\right ) \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (48 \textit {\_Z}^{3}+5 \textit {\_Z} +144\right )}{\sum }\frac {\left (31104 \textit {\_R1}^{2}-31079 \textit {\_R1} +2110\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-x +\textit {\_R1} \right )}{144 \textit {\_R1}^{2}+5}\right )}{6718589}+\frac {120 \ln \left (3\right ) \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (48 \textit {\_Z}^{3}+5 \textit {\_Z} +144\right )}{\sum }\frac {\left (25 \textit {\_R1}^{2}-1105 \textit {\_R1} -91152\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-x +\textit {\_R1} \right )}{144 \textit {\_R1}^{2}+5}\right )}{6718589}\)	\(259\)
orering	\(\frac {\left (5+x \right ) \left (\left (240 x^{4}-960 x^{3}+25 x^{2}+670 x -720\right ) \ln \left (3\right ) {\mathrm e}^{x}+2304 x^{8}+480 x^{6}+13824 x^{5}+25 x^{4}+1440 x^{3}+20736 x^{2}\right )}{2304 x^{8}+480 x^{6}+13824 x^{5}+25 x^{4}+1440 x^{3}+20736 x^{2}}-\frac {\left (48 x^{5}-955 x^{3}+154 x^{2}+382 x -720\right ) x \left (48 x^{3}+5 x +144\right ) \left (\frac {\left (960 x^{3}-2880 x^{2}+50 x +670\right ) \ln \left (3\right ) {\mathrm e}^{x}+\left (240 x^{4}-960 x^{3}+25 x^{2}+670 x -720\right ) \ln \left (3\right ) {\mathrm e}^{x}+18432 x^{7}+2880 x^{5}+69120 x^{4}+100 x^{3}+4320 x^{2}+41472 x}{2304 x^{8}+480 x^{6}+13824 x^{5}+25 x^{4}+1440 x^{3}+20736 x^{2}}-\frac {\left (\left (240 x^{4}-960 x^{3}+25 x^{2}+670 x -720\right ) \ln \left (3\right ) {\mathrm e}^{x}+2304 x^{8}+480 x^{6}+13824 x^{5}+25 x^{4}+1440 x^{3}+20736 x^{2}\right ) \left (18432 x^{7}+2880 x^{5}+69120 x^{4}+100 x^{3}+4320 x^{2}+41472 x \right )}{\left (2304 x^{8}+480 x^{6}+13824 x^{5}+25 x^{4}+1440 x^{3}+20736 x^{2}\right )^{2}}\right )}{2304 x^{8}-18432 x^{7}+46560 x^{6}+10944 x^{5}-64775 x^{4}+28988 x^{3}+16566 x^{2}-37152 x +41472}\)	\(399\)
default	\(\text {Expression too large to display}\)	\(722\)

Input:

int(((240*x^4-960*x^3+25*x^2+670*x-720)*ln(3)*exp(x)+2304*x^8+480*x^6+1382 
4*x^5+25*x^4+1440*x^3+20736*x^2)/(2304*x^8+480*x^6+13824*x^5+25*x^4+1440*x 
^3+20736*x^2),x,method=_RETURNVERBOSE)
 

Output:

x+5*ln(3)/x/(48*x^3+5*x+144)*exp(x)
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=\frac {48 \, x^{5} + 5 \, x^{3} + 144 \, x^{2} + 5 \, e^{x} \log \left (3\right )}{48 \, x^{4} + 5 \, x^{2} + 144 \, x} \] Input:

integrate(((240*x^4-960*x^3+25*x^2+670*x-720)*log(3)*exp(x)+2304*x^8+480*x 
^6+13824*x^5+25*x^4+1440*x^3+20736*x^2)/(2304*x^8+480*x^6+13824*x^5+25*x^4 
+1440*x^3+20736*x^2),x, algorithm="fricas")
 

Output:

(48*x^5 + 5*x^3 + 144*x^2 + 5*e^x*log(3))/(48*x^4 + 5*x^2 + 144*x)
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=x + \frac {5 e^{x} \log {\left (3 \right )}}{48 x^{4} + 5 x^{2} + 144 x} \] Input:

integrate(((240*x**4-960*x**3+25*x**2+670*x-720)*ln(3)*exp(x)+2304*x**8+48 
0*x**6+13824*x**5+25*x**4+1440*x**3+20736*x**2)/(2304*x**8+480*x**6+13824* 
x**5+25*x**4+1440*x**3+20736*x**2),x)
 

Output:

x + 5*exp(x)*log(3)/(48*x**4 + 5*x**2 + 144*x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=\frac {48 \, x^{5} + 5 \, x^{3} + 144 \, x^{2} + 5 \, e^{x} \log \left (3\right )}{48 \, x^{4} + 5 \, x^{2} + 144 \, x} \] Input:

integrate(((240*x^4-960*x^3+25*x^2+670*x-720)*log(3)*exp(x)+2304*x^8+480*x 
^6+13824*x^5+25*x^4+1440*x^3+20736*x^2)/(2304*x^8+480*x^6+13824*x^5+25*x^4 
+1440*x^3+20736*x^2),x, algorithm="maxima")
 

Output:

(48*x^5 + 5*x^3 + 144*x^2 + 5*e^x*log(3))/(48*x^4 + 5*x^2 + 144*x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=\frac {48 \, x^{5} + 5 \, x^{3} + 144 \, x^{2} + 5 \, e^{x} \log \left (3\right )}{48 \, x^{4} + 5 \, x^{2} + 144 \, x} \] Input:

integrate(((240*x^4-960*x^3+25*x^2+670*x-720)*log(3)*exp(x)+2304*x^8+480*x 
^6+13824*x^5+25*x^4+1440*x^3+20736*x^2)/(2304*x^8+480*x^6+13824*x^5+25*x^4 
+1440*x^3+20736*x^2),x, algorithm="giac")
 

Output:

(48*x^5 + 5*x^3 + 144*x^2 + 5*e^x*log(3))/(48*x^4 + 5*x^2 + 144*x)
 

Mupad [B] (verification not implemented)

Time = 2.72 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=x+\frac {5\,{\mathrm {e}}^x\,\ln \left (3\right )}{x\,\left (48\,x^3+5\,x+144\right )} \] Input:

int((20736*x^2 + 1440*x^3 + 25*x^4 + 13824*x^5 + 480*x^6 + 2304*x^8 + exp( 
x)*log(3)*(670*x + 25*x^2 - 960*x^3 + 240*x^4 - 720))/(20736*x^2 + 1440*x^ 
3 + 25*x^4 + 13824*x^5 + 480*x^6 + 2304*x^8),x)
 

Output:

x + (5*exp(x)*log(3))/(x*(5*x + 48*x^3 + 144))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8+e^x \left (-720+670 x+25 x^2-960 x^3+240 x^4\right ) \log (3)}{20736 x^2+1440 x^3+25 x^4+13824 x^5+480 x^6+2304 x^8} \, dx=\frac {5 e^{x} \mathrm {log}\left (3\right )+48 x^{5}+5 x^{3}+144 x^{2}}{x \left (48 x^{3}+5 x +144\right )} \] Input:

int(((240*x^4-960*x^3+25*x^2+670*x-720)*log(3)*exp(x)+2304*x^8+480*x^6+138 
24*x^5+25*x^4+1440*x^3+20736*x^2)/(2304*x^8+480*x^6+13824*x^5+25*x^4+1440* 
x^3+20736*x^2),x)
 

Output:

(5*e**x*log(3) + 48*x**5 + 5*x**3 + 144*x**2)/(x*(48*x**3 + 5*x + 144))