\(\int \frac {75 x^3-30 x^4+3 x^5+e^{3/x} (15+2 x+24 x^2-10 x^3+x^4)+(-e^{3/x} x^2-3 x^3) \log (\frac {e^{3/x}+3 x}{x})}{75 x^5-30 x^6+3 x^7+e^{3/x} (25 x^4-10 x^5+x^6)+(-30 x^4+6 x^5+e^{3/x} (-10 x^3+2 x^4)) \log (\frac {e^{3/x}+3 x}{x})+(e^{3/x} x^2+3 x^3) \log ^2(\frac {e^{3/x}+3 x}{x})} \, dx\) [2498]

Optimal result

Integrand size = 201, antiderivative size = 30 \[ \int \frac {75 x^3-30 x^4+3 x^5+e^{3/x} \left (15+2 x+24 x^2-10 x^3+x^4\right )+\left (-e^{3/x} x^2-3 x^3\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )}{75 x^5-30 x^6+3 x^7+e^{3/x} \left (25 x^4-10 x^5+x^6\right )+\left (-30 x^4+6 x^5+e^{3/x} \left (-10 x^3+2 x^4\right )\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )+\left (e^{3/x} x^2+3 x^3\right ) \log ^2\left (\frac {e^{3/x}+3 x}{x}\right )} \, dx=4+\frac {1}{-x+\frac {\log \left (3+\frac {e^{3/x}}{x}\right )}{5-x}} \] Output:

1/(ln(3+exp(3/x)/x)/(5-x)-x)+4
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {75 x^3-30 x^4+3 x^5+e^{3/x} \left (15+2 x+24 x^2-10 x^3+x^4\right )+\left (-e^{3/x} x^2-3 x^3\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )}{75 x^5-30 x^6+3 x^7+e^{3/x} \left (25 x^4-10 x^5+x^6\right )+\left (-30 x^4+6 x^5+e^{3/x} \left (-10 x^3+2 x^4\right )\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )+\left (e^{3/x} x^2+3 x^3\right ) \log ^2\left (\frac {e^{3/x}+3 x}{x}\right )} \, dx=\frac {5-x}{-5 x+x^2+\log \left (3+\frac {e^{3/x}}{x}\right )} \] Input:

Integrate[(75*x^3 - 30*x^4 + 3*x^5 + E^(3/x)*(15 + 2*x + 24*x^2 - 10*x^3 + 
 x^4) + (-(E^(3/x)*x^2) - 3*x^3)*Log[(E^(3/x) + 3*x)/x])/(75*x^5 - 30*x^6 
+ 3*x^7 + E^(3/x)*(25*x^4 - 10*x^5 + x^6) + (-30*x^4 + 6*x^5 + E^(3/x)*(-1 
0*x^3 + 2*x^4))*Log[(E^(3/x) + 3*x)/x] + (E^(3/x)*x^2 + 3*x^3)*Log[(E^(3/x 
) + 3*x)/x]^2),x]
 

Output:

(5 - x)/(-5*x + x^2 + Log[3 + E^(3/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.

\(\displaystyle \int \frac {3 x^5-30 x^4+75 x^3+\left (-3 x^3-e^{3/x} x^2\right ) \log \left (\frac {3 x+e^{3/x}}{x}\right )+e^{3/x} \left (x^4-10 x^3+24 x^2+2 x+15\right )}{3 x^7-30 x^6+75 x^5+\left (3 x^3+e^{3/x} x^2\right ) \log ^2\left (\frac {3 x+e^{3/x}}{x}\right )+e^{3/x} \left (x^6-10 x^5+25 x^4\right )+\left (6 x^5-30 x^4+e^{3/x} \left (2 x^4-10 x^3\right )\right ) \log \left (\frac {3 x+e^{3/x}}{x}\right )} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (x^2-2 x-15\right )}{x \left (3 x+e^{3/x}\right ) \left (x^2-5 x+\log \left (\frac {e^{3/x}}{x}+3\right )\right )^2}+\frac {x^4-10 x^3+24 x^2-x^2 \log \left (\frac {e^{3/x}}{x}+3\right )+2 x+15}{x^2 \left (x^2-5 x+\log \left (\frac {e^{3/x}}{x}+3\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 24 \int \frac {1}{\left (x^2-5 x+\log \left (3+\frac {e^{3/x}}{x}\right )\right )^2}dx+15 \int \frac {1}{x^2 \left (x^2-5 x+\log \left (3+\frac {e^{3/x}}{x}\right )\right )^2}dx+2 \int \frac {1}{x \left (x^2-5 x+\log \left (3+\frac {e^{3/x}}{x}\right )\right )^2}dx-15 \int \frac {x}{\left (x^2-5 x+\log \left (3+\frac {e^{3/x}}{x}\right )\right )^2}dx+2 \int \frac {x^2}{\left (x^2-5 x+\log \left (3+\frac {e^{3/x}}{x}\right )\right )^2}dx-6 \int \frac {1}{\left (3 x+e^{3/x}\right ) \left (x^2-5 x+\log \left (3+\frac {e^{3/x}}{x}\right )\right )^2}dx-45 \int \frac {1}{x \left (3 x+e^{3/x}\right ) \left (x^2-5 x+\log \left (3+\frac {e^{3/x}}{x}\right )\right )^2}dx+3 \int \frac {x}{\left (3 x+e^{3/x}\right ) \left (x^2-5 x+\log \left (3+\frac {e^{3/x}}{x}\right )\right )^2}dx-\int \frac {1}{x^2-5 x+\log \left (3+\frac {e^{3/x}}{x}\right )}dx\)

Input:

Int[(75*x^3 - 30*x^4 + 3*x^5 + E^(3/x)*(15 + 2*x + 24*x^2 - 10*x^3 + x^4) 
+ (-(E^(3/x)*x^2) - 3*x^3)*Log[(E^(3/x) + 3*x)/x])/(75*x^5 - 30*x^6 + 3*x^ 
7 + E^(3/x)*(25*x^4 - 10*x^5 + x^6) + (-30*x^4 + 6*x^5 + E^(3/x)*(-10*x^3 
+ 2*x^4))*Log[(E^(3/x) + 3*x)/x] + (E^(3/x)*x^2 + 3*x^3)*Log[(E^(3/x) + 3* 
x)/x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.50 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\(\frac {30-6 x}{6 x^{2}+6 \ln \left (\frac {{\mathrm e}^{\frac {3}{x}}+3 x}{x}\right )-30 x}\)	\(32\)
risch	\(-\frac {2 \left (-5+x \right )}{i \pi \,\operatorname {csgn}\left (i \left (\frac {{\mathrm e}^{\frac {3}{x}}}{3}+x \right )\right ) {\operatorname {csgn}\left (\frac {i \left (\frac {{\mathrm e}^{\frac {3}{x}}}{3}+x \right )}{x}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (\frac {{\mathrm e}^{\frac {3}{x}}}{3}+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {{\mathrm e}^{\frac {3}{x}}}{3}+x \right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (\frac {{\mathrm e}^{\frac {3}{x}}}{3}+x \right )}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-i \pi {\operatorname {csgn}\left (\frac {i \left (\frac {{\mathrm e}^{\frac {3}{x}}}{3}+x \right )}{x}\right )}^{3}+2 x^{2}+2 \ln \left (3\right )-10 x -2 \ln \left (x \right )+2 \ln \left (\frac {{\mathrm e}^{\frac {3}{x}}}{3}+x \right )}\)	\(170\)

Input:

int(((-x^2*exp(3/x)-3*x^3)*ln((exp(3/x)+3*x)/x)+(x^4-10*x^3+24*x^2+2*x+15) 
*exp(3/x)+3*x^5-30*x^4+75*x^3)/((x^2*exp(3/x)+3*x^3)*ln((exp(3/x)+3*x)/x)^ 
2+((2*x^4-10*x^3)*exp(3/x)+6*x^5-30*x^4)*ln((exp(3/x)+3*x)/x)+(x^6-10*x^5+ 
25*x^4)*exp(3/x)+3*x^7-30*x^6+75*x^5),x,method=_RETURNVERBOSE)
 

Output:

1/6*(30-6*x)/(x^2+ln((exp(3/x)+3*x)/x)-5*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {75 x^3-30 x^4+3 x^5+e^{3/x} \left (15+2 x+24 x^2-10 x^3+x^4\right )+\left (-e^{3/x} x^2-3 x^3\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )}{75 x^5-30 x^6+3 x^7+e^{3/x} \left (25 x^4-10 x^5+x^6\right )+\left (-30 x^4+6 x^5+e^{3/x} \left (-10 x^3+2 x^4\right )\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )+\left (e^{3/x} x^2+3 x^3\right ) \log ^2\left (\frac {e^{3/x}+3 x}{x}\right )} \, dx=-\frac {x - 5}{x^{2} - 5 \, x + \log \left (\frac {3 \, x + e^{\frac {3}{x}}}{x}\right )} \] Input:

integrate(((-x^2*exp(3/x)-3*x^3)*log((exp(3/x)+3*x)/x)+(x^4-10*x^3+24*x^2+ 
2*x+15)*exp(3/x)+3*x^5-30*x^4+75*x^3)/((x^2*exp(3/x)+3*x^3)*log((exp(3/x)+ 
3*x)/x)^2+((2*x^4-10*x^3)*exp(3/x)+6*x^5-30*x^4)*log((exp(3/x)+3*x)/x)+(x^ 
6-10*x^5+25*x^4)*exp(3/x)+3*x^7-30*x^6+75*x^5),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {75 x^3-30 x^4+3 x^5+e^{3/x} \left (15+2 x+24 x^2-10 x^3+x^4\right )+\left (-e^{3/x} x^2-3 x^3\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )}{75 x^5-30 x^6+3 x^7+e^{3/x} \left (25 x^4-10 x^5+x^6\right )+\left (-30 x^4+6 x^5+e^{3/x} \left (-10 x^3+2 x^4\right )\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )+\left (e^{3/x} x^2+3 x^3\right ) \log ^2\left (\frac {e^{3/x}+3 x}{x}\right )} \, dx=\frac {5 - x}{x^{2} - 5 x + \log {\left (\frac {3 x + e^{\frac {3}{x}}}{x} \right )}} \] Input:

integrate(((-x**2*exp(3/x)-3*x**3)*ln((exp(3/x)+3*x)/x)+(x**4-10*x**3+24*x 
**2+2*x+15)*exp(3/x)+3*x**5-30*x**4+75*x**3)/((x**2*exp(3/x)+3*x**3)*ln((e 
xp(3/x)+3*x)/x)**2+((2*x**4-10*x**3)*exp(3/x)+6*x**5-30*x**4)*ln((exp(3/x) 
+3*x)/x)+(x**6-10*x**5+25*x**4)*exp(3/x)+3*x**7-30*x**6+75*x**5),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {75 x^3-30 x^4+3 x^5+e^{3/x} \left (15+2 x+24 x^2-10 x^3+x^4\right )+\left (-e^{3/x} x^2-3 x^3\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )}{75 x^5-30 x^6+3 x^7+e^{3/x} \left (25 x^4-10 x^5+x^6\right )+\left (-30 x^4+6 x^5+e^{3/x} \left (-10 x^3+2 x^4\right )\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )+\left (e^{3/x} x^2+3 x^3\right ) \log ^2\left (\frac {e^{3/x}+3 x}{x}\right )} \, dx=-\frac {x - 5}{x^{2} - 5 \, x + \log \left (3 \, x + e^{\frac {3}{x}}\right ) - \log \left (x\right )} \] Input:

integrate(((-x^2*exp(3/x)-3*x^3)*log((exp(3/x)+3*x)/x)+(x^4-10*x^3+24*x^2+ 
2*x+15)*exp(3/x)+3*x^5-30*x^4+75*x^3)/((x^2*exp(3/x)+3*x^3)*log((exp(3/x)+ 
3*x)/x)^2+((2*x^4-10*x^3)*exp(3/x)+6*x^5-30*x^4)*log((exp(3/x)+3*x)/x)+(x^ 
6-10*x^5+25*x^4)*exp(3/x)+3*x^7-30*x^6+75*x^5),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {75 x^3-30 x^4+3 x^5+e^{3/x} \left (15+2 x+24 x^2-10 x^3+x^4\right )+\left (-e^{3/x} x^2-3 x^3\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )}{75 x^5-30 x^6+3 x^7+e^{3/x} \left (25 x^4-10 x^5+x^6\right )+\left (-30 x^4+6 x^5+e^{3/x} \left (-10 x^3+2 x^4\right )\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )+\left (e^{3/x} x^2+3 x^3\right ) \log ^2\left (\frac {e^{3/x}+3 x}{x}\right )} \, dx=-\frac {x - 5}{x^{2} - 5 \, x + \log \left (\frac {3 \, x + e^{\frac {3}{x}}}{x}\right )} \] Input:

integrate(((-x^2*exp(3/x)-3*x^3)*log((exp(3/x)+3*x)/x)+(x^4-10*x^3+24*x^2+ 
2*x+15)*exp(3/x)+3*x^5-30*x^4+75*x^3)/((x^2*exp(3/x)+3*x^3)*log((exp(3/x)+ 
3*x)/x)^2+((2*x^4-10*x^3)*exp(3/x)+6*x^5-30*x^4)*log((exp(3/x)+3*x)/x)+(x^ 
6-10*x^5+25*x^4)*exp(3/x)+3*x^7-30*x^6+75*x^5),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {75 x^3-30 x^4+3 x^5+e^{3/x} \left (15+2 x+24 x^2-10 x^3+x^4\right )+\left (-e^{3/x} x^2-3 x^3\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )}{75 x^5-30 x^6+3 x^7+e^{3/x} \left (25 x^4-10 x^5+x^6\right )+\left (-30 x^4+6 x^5+e^{3/x} \left (-10 x^3+2 x^4\right )\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )+\left (e^{3/x} x^2+3 x^3\right ) \log ^2\left (\frac {e^{3/x}+3 x}{x}\right )} \, dx=\int \frac {{\mathrm {e}}^{3/x}\,\left (x^4-10\,x^3+24\,x^2+2\,x+15\right )-\ln \left (\frac {3\,x+{\mathrm {e}}^{3/x}}{x}\right )\,\left (x^2\,{\mathrm {e}}^{3/x}+3\,x^3\right )+75\,x^3-30\,x^4+3\,x^5}{{\ln \left (\frac {3\,x+{\mathrm {e}}^{3/x}}{x}\right )}^2\,\left (x^2\,{\mathrm {e}}^{3/x}+3\,x^3\right )-\ln \left (\frac {3\,x+{\mathrm {e}}^{3/x}}{x}\right )\,\left ({\mathrm {e}}^{3/x}\,\left (10\,x^3-2\,x^4\right )+30\,x^4-6\,x^5\right )+{\mathrm {e}}^{3/x}\,\left (x^6-10\,x^5+25\,x^4\right )+75\,x^5-30\,x^6+3\,x^7} \,d x \] Input:

int((exp(3/x)*(2*x + 24*x^2 - 10*x^3 + x^4 + 15) - log((3*x + exp(3/x))/x) 
*(x^2*exp(3/x) + 3*x^3) + 75*x^3 - 30*x^4 + 3*x^5)/(log((3*x + exp(3/x))/x 
)^2*(x^2*exp(3/x) + 3*x^3) - log((3*x + exp(3/x))/x)*(exp(3/x)*(10*x^3 - 2 
*x^4) + 30*x^4 - 6*x^5) + exp(3/x)*(25*x^4 - 10*x^5 + x^6) + 75*x^5 - 30*x 
^6 + 3*x^7),x)
 

Output:

int((exp(3/x)*(2*x + 24*x^2 - 10*x^3 + x^4 + 15) - log((3*x + exp(3/x))/x) 
*(x^2*exp(3/x) + 3*x^3) + 75*x^3 - 30*x^4 + 3*x^5)/(log((3*x + exp(3/x))/x 
)^2*(x^2*exp(3/x) + 3*x^3) - log((3*x + exp(3/x))/x)*(exp(3/x)*(10*x^3 - 2 
*x^4) + 30*x^4 - 6*x^5) + exp(3/x)*(25*x^4 - 10*x^5 + x^6) + 75*x^5 - 30*x 
^6 + 3*x^7), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {75 x^3-30 x^4+3 x^5+e^{3/x} \left (15+2 x+24 x^2-10 x^3+x^4\right )+\left (-e^{3/x} x^2-3 x^3\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )}{75 x^5-30 x^6+3 x^7+e^{3/x} \left (25 x^4-10 x^5+x^6\right )+\left (-30 x^4+6 x^5+e^{3/x} \left (-10 x^3+2 x^4\right )\right ) \log \left (\frac {e^{3/x}+3 x}{x}\right )+\left (e^{3/x} x^2+3 x^3\right ) \log ^2\left (\frac {e^{3/x}+3 x}{x}\right )} \, dx=\frac {-x +5}{\mathrm {log}\left (\frac {e^{\frac {3}{x}}+3 x}{x}\right )+x^{2}-5 x} \] Input:

int(((-x^2*exp(3/x)-3*x^3)*log((exp(3/x)+3*x)/x)+(x^4-10*x^3+24*x^2+2*x+15 
)*exp(3/x)+3*x^5-30*x^4+75*x^3)/((x^2*exp(3/x)+3*x^3)*log((exp(3/x)+3*x)/x 
)^2+((2*x^4-10*x^3)*exp(3/x)+6*x^5-30*x^4)*log((exp(3/x)+3*x)/x)+(x^6-10*x 
^5+25*x^4)*exp(3/x)+3*x^7-30*x^6+75*x^5),x)
 

Output:

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