\(\int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} (18-\log (4 x+2 x^2))}-x} (-6 x-3 x^2+e^{\frac {1}{6} (18-\log (4 x+2 x^2))} (1+x))}{6 x+3 x^2} \, dx\) [736]

Optimal result

Integrand size = 84, antiderivative size = 27 \[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=e^{5-x-\frac {e^3}{\sqrt [6]{x (4+2 x)}}}-x \] Output:

exp(5-exp(3-1/6*ln(x*(4+2*x)))-x)-x
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=\int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx \] Input:

Integrate[(-6*x - 3*x^2 + E^(5 - E^((18 - Log[4*x + 2*x^2])/6) - x)*(-6*x 
- 3*x^2 + E^((18 - Log[4*x + 2*x^2])/6)*(1 + x)))/(6*x + 3*x^2),x]
 

Output:

Integrate[(-6*x - 3*x^2 + E^(5 - E^((18 - Log[4*x + 2*x^2])/6) - x)*(-6*x 
- 3*x^2 + E^((18 - Log[4*x + 2*x^2])/6)*(1 + x)))/(6*x + 3*x^2), 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.

\(\displaystyle \int \frac {-3 x^2+e^{-e^{\frac {1}{6} \left (18-\log \left (2 x^2+4 x\right )\right )}-x+5} \left (-3 x^2+(x+1) e^{\frac {1}{6} \left (18-\log \left (2 x^2+4 x\right )\right )}-6 x\right )-6 x}{3 x^2+6 x} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {-3 x^2+e^{-e^{\frac {1}{6} \left (18-\log \left (2 x^2+4 x\right )\right )}-x+5} \left (-3 x^2+(x+1) e^{\frac {1}{6} \left (18-\log \left (2 x^2+4 x\right )\right )}-6 x\right )-6 x}{x (3 x+6)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (x+2)}}+8} (x+1)}{3 \sqrt [6]{2} x (x+2) \sqrt [6]{x^2+2 x}}-e^{-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (x+2)}}} \left (e^{x+\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (x+2)}}}+e^5\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2^{5/6} \sqrt [6]{x+2} \sqrt [6]{x} \text {Subst}\left (\int \frac {e^{-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (x^6+2\right )}}+8}}{x^2 \left (x^6+2\right )^{7/6}}dx,x,\sqrt [6]{x}\right )}{\sqrt [6]{x^2+2 x}}+\frac {2^{5/6} \sqrt [6]{x+2} \sqrt [6]{x} \text {Subst}\left (\int \frac {e^{-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (x^6+2\right )}}+8} x^4}{\left (x^6+2\right )^{7/6}}dx,x,\sqrt [6]{x}\right )}{\sqrt [6]{x^2+2 x}}-\int e^{-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (x+2)}}+5}dx-x\)

Input:

Int[(-6*x - 3*x^2 + E^(5 - E^((18 - Log[4*x + 2*x^2])/6) - x)*(-6*x - 3*x^ 
2 + E^((18 - Log[4*x + 2*x^2])/6)*(1 + x)))/(6*x + 3*x^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\(1-x +{\mathrm e}^{-{\mathrm e}^{-\frac {\ln \left (2 x^{2}+4 x \right )}{6}+3}+5-x}\)	\(29\)

Input:

int((((1+x)*exp(-1/6*ln(2*x^2+4*x)+3)-3*x^2-6*x)*exp(-exp(-1/6*ln(2*x^2+4* 
x)+3)+5-x)-3*x^2-6*x)/(3*x^2+6*x),x,method=_RETURNVERBOSE)
 

Output:

1-x+exp(-exp(-1/6*ln(2*x^2+4*x)+3)+5-x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=-x + e^{\left (-\frac {2 \, x^{3} - 6 \, x^{2} + {\left (2 \, x^{2} + 4 \, x\right )}^{\frac {5}{6}} e^{3} - 20 \, x}{2 \, {\left (x^{2} + 2 \, x\right )}}\right )} \] Input:

integrate((((1+x)*exp(-1/6*log(2*x^2+4*x)+3)-3*x^2-6*x)*exp(-exp(-1/6*log( 
2*x^2+4*x)+3)+5-x)-3*x^2-6*x)/(3*x^2+6*x),x, algorithm="fricas")
 

Output:

-x + e^(-1/2*(2*x^3 - 6*x^2 + (2*x^2 + 4*x)^(5/6)*e^3 - 20*x)/(x^2 + 2*x))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=\text {Timed out} \] Input:

integrate((((1+x)*exp(-1/6*ln(2*x**2+4*x)+3)-3*x**2-6*x)*exp(-exp(-1/6*ln( 
2*x**2+4*x)+3)+5-x)-3*x**2-6*x)/(3*x**2+6*x),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=-x + e^{\left (-\frac {2^{\frac {5}{6}} e^{3}}{2 \, {\left (x + 2\right )}^{\frac {1}{6}} x^{\frac {1}{6}}} - x + 5\right )} \] Input:

integrate((((1+x)*exp(-1/6*log(2*x^2+4*x)+3)-3*x^2-6*x)*exp(-exp(-1/6*log( 
2*x^2+4*x)+3)+5-x)-3*x^2-6*x)/(3*x^2+6*x),x, algorithm="maxima")
 

Output:

-x + e^(-1/2*2^(5/6)*e^3/((x + 2)^(1/6)*x^(1/6)) - x + 5)
 

Giac [F]

\[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=\int { -\frac {3 \, x^{2} + {\left (3 \, x^{2} - {\left (x + 1\right )} e^{\left (-\frac {1}{6} \, \log \left (2 \, x^{2} + 4 \, x\right ) + 3\right )} + 6 \, x\right )} e^{\left (-x - e^{\left (-\frac {1}{6} \, \log \left (2 \, x^{2} + 4 \, x\right ) + 3\right )} + 5\right )} + 6 \, x}{3 \, {\left (x^{2} + 2 \, x\right )}} \,d x } \] Input:

integrate((((1+x)*exp(-1/6*log(2*x^2+4*x)+3)-3*x^2-6*x)*exp(-exp(-1/6*log( 
2*x^2+4*x)+3)+5-x)-3*x^2-6*x)/(3*x^2+6*x),x, algorithm="giac")
 

Output:

integrate(-1/3*(3*x^2 + (3*x^2 - (x + 1)*e^(-1/6*log(2*x^2 + 4*x) + 3) + 6 
*x)*e^(-x - e^(-1/6*log(2*x^2 + 4*x) + 3) + 5) + 6*x)/(x^2 + 2*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=\int -\frac {6\,x+3\,x^2+{\mathrm {e}}^{5-{\mathrm {e}}^{3-\frac {\ln \left (2\,x^2+4\,x\right )}{6}}-x}\,\left (6\,x-{\mathrm {e}}^{3-\frac {\ln \left (2\,x^2+4\,x\right )}{6}}\,\left (x+1\right )+3\,x^2\right )}{3\,x^2+6\,x} \,d x \] Input:

int(-(6*x + 3*x^2 + exp(5 - exp(3 - log(4*x + 2*x^2)/6) - x)*(6*x - exp(3 
- log(4*x + 2*x^2)/6)*(x + 1) + 3*x^2))/(6*x + 3*x^2),x)
 

Output:

int(-(6*x + 3*x^2 + exp(5 - exp(3 - log(4*x + 2*x^2)/6) - x)*(6*x - exp(3 
- log(4*x + 2*x^2)/6)*(x + 1) + 3*x^2))/(6*x + 3*x^2), x)
 

Reduce [F]

\[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=\int \frac {\left (\left (x +1\right ) {\mathrm e}^{-\frac {\mathrm {log}\left (2 x^{2}+4 x \right )}{6}+3}-3 x^{2}-6 x \right ) {\mathrm e}^{-{\mathrm e}^{-\frac {\mathrm {log}\left (2 x^{2}+4 x \right )}{6}+3}+5-x}-3 x^{2}-6 x}{3 x^{2}+6 x}d x \] Input:

int((((1+x)*exp(-1/6*log(2*x^2+4*x)+3)-3*x^2-6*x)*exp(-exp(-1/6*log(2*x^2+ 
4*x)+3)+5-x)-3*x^2-6*x)/(3*x^2+6*x),x)
 

Output:

int((((1+x)*exp(-1/6*log(2*x^2+4*x)+3)-3*x^2-6*x)*exp(-exp(-1/6*log(2*x^2+ 
4*x)+3)+5-x)-3*x^2-6*x)/(3*x^2+6*x),x)