\(\int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+(-32 x-16 x^2-2 x^3) \log (\frac {4 e^{\frac {14+3 x}{4+x}}}{x})}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+(-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5) \log (\frac {4 e^{\frac {14+3 x}{4+x}}}{x})+(16+8 x+x^2) \log ^2(\frac {4 e^{\frac {14+3 x}{4+x}}}{x})} \, dx\) [2032]

Optimal result

Integrand size = 177, antiderivative size = 32 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=\frac {x^2}{5-x^3-\log \left (\frac {4 e^{3+\frac {2}{4+x}}}{x}\right )} \] Output:

x^2/(5-ln(4*exp(3+2/(4+x))/x)-x^3)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=-\frac {x^2}{-5+x^3+\log \left (\frac {4 e^{3+\frac {2}{4+x}}}{x}\right )} \] Input:

Integrate[(144*x + 70*x^2 + 9*x^3 + 16*x^4 + 8*x^5 + x^6 + (-32*x - 16*x^2 
 - 2*x^3)*Log[(4*E^((14 + 3*x)/(4 + x)))/x])/(400 + 200*x + 25*x^2 - 160*x 
^3 - 80*x^4 - 10*x^5 + 16*x^6 + 8*x^7 + x^8 + (-160 - 80*x - 10*x^2 + 32*x 
^3 + 16*x^4 + 2*x^5)*Log[(4*E^((14 + 3*x)/(4 + x)))/x] + (16 + 8*x + x^2)* 
Log[(4*E^((14 + 3*x)/(4 + x)))/x]^2),x]
 

Output:

-(x^2/(-5 + x^3 + Log[(4*E^(3 + 2/(4 + 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.

\(\displaystyle \int \frac {x^6+8 x^5+16 x^4+9 x^3+70 x^2+\left (-2 x^3-16 x^2-32 x\right ) \log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )+144 x}{x^8+8 x^7+16 x^6-10 x^5-80 x^4-160 x^3+25 x^2+\left (x^2+8 x+16\right ) \log ^2\left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )+\left (2 x^5+16 x^4+32 x^3-10 x^2-80 x-160\right ) \log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )+200 x+400} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (x^5+8 x^4+16 x^3+9 x^2+70 x-2 (x+4)^2 \log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )+144\right )}{(x+4)^2 \left (-x^3-\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x \left (3 x^5+24 x^4+48 x^3-x^2-10 x-16\right )}{(x+4)^2 \left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}-\frac {2 x}{x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int \frac {1}{\left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}dx-\int \frac {x}{\left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}dx-32 \int \frac {1}{(x+4)^2 \left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}dx+16 \int \frac {1}{(x+4) \left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}dx-2 \int \frac {x}{x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5}dx+3 \int \frac {x^4}{\left (x^3+\log \left (\frac {4 e^{\frac {3 x+14}{x+4}}}{x}\right )-5\right )^2}dx\)

Input:

Int[(144*x + 70*x^2 + 9*x^3 + 16*x^4 + 8*x^5 + x^6 + (-32*x - 16*x^2 - 2*x 
^3)*Log[(4*E^((14 + 3*x)/(4 + x)))/x])/(400 + 200*x + 25*x^2 - 160*x^3 - 8 
0*x^4 - 10*x^5 + 16*x^6 + 8*x^7 + x^8 + (-160 - 80*x - 10*x^2 + 32*x^3 + 1 
6*x^4 + 2*x^5)*Log[(4*E^((14 + 3*x)/(4 + x)))/x] + (16 + 8*x + x^2)*Log[(4 
*E^((14 + 3*x)/(4 + x)))/x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.78 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38

method	result	size
parallelrisch	\(-\frac {65632 x^{3}+262528 x^{2}}{65632 \left (x^{3}+\ln \left (\frac {4 \,{\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )-5\right ) \left (4+x \right )}\)	\(44\)
default	\(\frac {x^{3}+4 x^{2}}{-x^{4}-4 x^{3}+x \ln \left (x \right )-\left (\ln \left (\frac {4 \,{\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )-\frac {3 x +14}{4+x}+\ln \left (x \right )\right ) x +2 x -4 \ln \left (\frac {4 \,{\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )+\frac {12 x +56}{4+x}+6}\)	\(100\)
risch	\(-\frac {2 x^{2}}{-i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\frac {3 x +14}{4+x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )^{2}+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{\frac {3 x +14}{4+x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )^{2}-10-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )^{3}+2 x^{3}+4 \ln \left (2\right )-2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{\frac {3 x +14}{4+x}}\right )}\)	\(182\)

Input:

int(((-2*x^3-16*x^2-32*x)*ln(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16*x^4+9*x 
^3+70*x^2+144*x)/((x^2+8*x+16)*ln(4*exp((3*x+14)/(4+x))/x)^2+(2*x^5+16*x^4 
+32*x^3-10*x^2-80*x-160)*ln(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+16*x^6-10*x 
^5-80*x^4-160*x^3+25*x^2+200*x+400),x,method=_RETURNVERBOSE)
 

Output:

-1/65632*(65632*x^3+262528*x^2)/(x^3+ln(4*exp((3*x+14)/(4+x))/x)-5)/(4+x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=-\frac {x^{2}}{x^{3} + \log \left (\frac {4 \, e^{\left (\frac {3 \, x + 14}{x + 4}\right )}}{x}\right ) - 5} \] Input:

integrate(((-2*x^3-16*x^2-32*x)*log(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16* 
x^4+9*x^3+70*x^2+144*x)/((x^2+8*x+16)*log(4*exp((3*x+14)/(4+x))/x)^2+(2*x^ 
5+16*x^4+32*x^3-10*x^2-80*x-160)*log(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+16 
*x^6-10*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x, algorithm="fricas")
 

Output:

-x^2/(x^3 + log(4*e^((3*x + 14)/(x + 4))/x) - 5)
 

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=- \frac {x^{2}}{x^{3} + \log {\left (\frac {4 e^{\frac {3 x + 14}{x + 4}}}{x} \right )} - 5} \] Input:

integrate(((-2*x**3-16*x**2-32*x)*ln(4*exp((3*x+14)/(4+x))/x)+x**6+8*x**5+ 
16*x**4+9*x**3+70*x**2+144*x)/((x**2+8*x+16)*ln(4*exp((3*x+14)/(4+x))/x)** 
2+(2*x**5+16*x**4+32*x**3-10*x**2-80*x-160)*ln(4*exp((3*x+14)/(4+x))/x)+x* 
*8+8*x**7+16*x**6-10*x**5-80*x**4-160*x**3+25*x**2+200*x+400),x)
 

Output:

-x**2/(x**3 + log(4*exp((3*x + 14)/(x + 4))/x) - 5)
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=-\frac {x^{3} + 4 \, x^{2}}{x^{4} + 4 \, x^{3} + 2 \, x {\left (\log \left (2\right ) - 1\right )} - {\left (x + 4\right )} \log \left (x\right ) + 8 \, \log \left (2\right ) - 6} \] Input:

integrate(((-2*x^3-16*x^2-32*x)*log(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16* 
x^4+9*x^3+70*x^2+144*x)/((x^2+8*x+16)*log(4*exp((3*x+14)/(4+x))/x)^2+(2*x^ 
5+16*x^4+32*x^3-10*x^2-80*x-160)*log(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+16 
*x^6-10*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x, algorithm="maxima")
 

Output:

-(x^3 + 4*x^2)/(x^4 + 4*x^3 + 2*x*(log(2) - 1) - (x + 4)*log(x) + 8*log(2) 
 - 6)
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=-\frac {x^{3} + 4 \, x^{2}}{x^{4} + 4 \, x^{3} + 2 \, x \log \left (2\right ) - x \log \left (x\right ) - 2 \, x + 8 \, \log \left (2\right ) - 4 \, \log \left (x\right ) - 6} \] Input:

integrate(((-2*x^3-16*x^2-32*x)*log(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16* 
x^4+9*x^3+70*x^2+144*x)/((x^2+8*x+16)*log(4*exp((3*x+14)/(4+x))/x)^2+(2*x^ 
5+16*x^4+32*x^3-10*x^2-80*x-160)*log(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+16 
*x^6-10*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x, algorithm="giac")
 

Output:

-(x^3 + 4*x^2)/(x^4 + 4*x^3 + 2*x*log(2) - x*log(x) - 2*x + 8*log(2) - 4*l 
og(x) - 6)
 

Mupad [B] (verification not implemented)

Time = 4.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=-\frac {x^2}{\ln \left (\frac {4}{x}\right )+\frac {3\,x}{x+4}+\frac {14}{x+4}+x^3-5} \] Input:

int((144*x + 70*x^2 + 9*x^3 + 16*x^4 + 8*x^5 + x^6 - log((4*exp((3*x + 14) 
/(x + 4)))/x)*(32*x + 16*x^2 + 2*x^3))/(200*x - log((4*exp((3*x + 14)/(x + 
 4)))/x)*(80*x + 10*x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 160) + log((4*exp((3*x 
 + 14)/(x + 4)))/x)^2*(8*x + x^2 + 16) + 25*x^2 - 160*x^3 - 80*x^4 - 10*x^ 
5 + 16*x^6 + 8*x^7 + x^8 + 400),x)
 

Output:

-x^2/(log(4/x) + (3*x)/(x + 4) + 14/(x + 4) + x^3 - 5)
 

Reduce [F]

\[ \int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx=\text {too large to display} \] Input:

int(((-2*x^3-16*x^2-32*x)*log(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16*x^4+9* 
x^3+70*x^2+144*x)/((x^2+8*x+16)*log(4*exp((3*x+14)/(4+x))/x)^2+(2*x^5+16*x 
^4+32*x^3-10*x^2-80*x-160)*log(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+16*x^6-1 
0*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x)
 

Output:

int(x**6/(log((4*e**(2/(x + 4))*e**3)/x)**2*x**2 + 8*log((4*e**(2/(x + 4)) 
*e**3)/x)**2*x + 16*log((4*e**(2/(x + 4))*e**3)/x)**2 + 2*log((4*e**(2/(x 
+ 4))*e**3)/x)*x**5 + 16*log((4*e**(2/(x + 4))*e**3)/x)*x**4 + 32*log((4*e 
**(2/(x + 4))*e**3)/x)*x**3 - 10*log((4*e**(2/(x + 4))*e**3)/x)*x**2 - 80* 
log((4*e**(2/(x + 4))*e**3)/x)*x - 160*log((4*e**(2/(x + 4))*e**3)/x) + x* 
*8 + 8*x**7 + 16*x**6 - 10*x**5 - 80*x**4 - 160*x**3 + 25*x**2 + 200*x + 4 
00),x) + 8*int(x**5/(log((4*e**(2/(x + 4))*e**3)/x)**2*x**2 + 8*log((4*e** 
(2/(x + 4))*e**3)/x)**2*x + 16*log((4*e**(2/(x + 4))*e**3)/x)**2 + 2*log(( 
4*e**(2/(x + 4))*e**3)/x)*x**5 + 16*log((4*e**(2/(x + 4))*e**3)/x)*x**4 + 
32*log((4*e**(2/(x + 4))*e**3)/x)*x**3 - 10*log((4*e**(2/(x + 4))*e**3)/x) 
*x**2 - 80*log((4*e**(2/(x + 4))*e**3)/x)*x - 160*log((4*e**(2/(x + 4))*e* 
*3)/x) + x**8 + 8*x**7 + 16*x**6 - 10*x**5 - 80*x**4 - 160*x**3 + 25*x**2 
+ 200*x + 400),x) + 16*int(x**4/(log((4*e**(2/(x + 4))*e**3)/x)**2*x**2 + 
8*log((4*e**(2/(x + 4))*e**3)/x)**2*x + 16*log((4*e**(2/(x + 4))*e**3)/x)* 
*2 + 2*log((4*e**(2/(x + 4))*e**3)/x)*x**5 + 16*log((4*e**(2/(x + 4))*e**3 
)/x)*x**4 + 32*log((4*e**(2/(x + 4))*e**3)/x)*x**3 - 10*log((4*e**(2/(x + 
4))*e**3)/x)*x**2 - 80*log((4*e**(2/(x + 4))*e**3)/x)*x - 160*log((4*e**(2 
/(x + 4))*e**3)/x) + x**8 + 8*x**7 + 16*x**6 - 10*x**5 - 80*x**4 - 160*x** 
3 + 25*x**2 + 200*x + 400),x) + 9*int(x**3/(log((4*e**(2/(x + 4))*e**3)/x) 
**2*x**2 + 8*log((4*e**(2/(x + 4))*e**3)/x)**2*x + 16*log((4*e**(2/(x +...