\(\int \frac {-x^2+2 x^3+e^{\frac {2 (30-40 x^2+x^3+x^2 \log (x))}{x}} (-60-78 x^2+4 x^3+2 x^2 \log (x))+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x))}{x^2} \, dx\) [371]

Optimal result

Integrand size = 107, antiderivative size = 29 \[ \int \frac {-x^2+2 x^3+e^{\frac {2 \left (30-40 x^2+x^3+x^2 \log (x)\right )}{x}} \left (-60-78 x^2+4 x^3+2 x^2 \log (x)\right )+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} \left (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x)\right )}{x^2} \, dx=-x+\left (e^{\frac {10 \left (3-4 x^2\right )}{x}+x (x+\log (x))}+x\right )^2 \] Output:

(x+exp(10/x*(-4*x^2+3)+(x+ln(x))*x))^2-x
 

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {-x^2+2 x^3+e^{\frac {2 \left (30-40 x^2+x^3+x^2 \log (x)\right )}{x}} \left (-60-78 x^2+4 x^3+2 x^2 \log (x)\right )+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} \left (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x)\right )}{x^2} \, dx=-x+x^2+e^{\frac {60}{x}-80 x+2 x^2} x^{2 x}+2 e^{\frac {30}{x}-40 x+x^2} x^{1+x} \] Input:

Integrate[(-x^2 + 2*x^3 + E^((2*(30 - 40*x^2 + x^3 + x^2*Log[x]))/x)*(-60 
- 78*x^2 + 4*x^3 + 2*x^2*Log[x]) + E^((30 - 40*x^2 + x^3 + x^2*Log[x])/x)* 
(-60*x + 2*x^2 - 78*x^3 + 4*x^4 + 2*x^3*Log[x]))/x^2,x]
 

Output:

-x + x^2 + E^(60/x - 80*x + 2*x^2)*x^(2*x) + 2*E^(30/x - 40*x + x^2)*x^(1 
+ 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.

\(\displaystyle \int \frac {2 x^3-x^2+e^{\frac {2 \left (x^3-40 x^2+x^2 \log (x)+30\right )}{x}} \left (4 x^3-78 x^2+2 x^2 \log (x)-60\right )+e^{\frac {x^3-40 x^2+x^2 \log (x)+30}{x}} \left (4 x^4-78 x^3+2 x^3 \log (x)+2 x^2-60 x\right )}{x^2} \, dx\)

\(\Big \downarrow \) 2010

\(\displaystyle \int \left (2 e^{x^2-40 x+\frac {30}{x}} x^{x-1} \left (2 x^3-39 x^2+x^2 \log (x)+x-30\right )+2 e^{\frac {2 \left (x^3-40 x^2+30\right )}{x}} x^{2 x-2} \left (2 x^3-39 x^2+x^2 \log (x)-30\right )+2 x-1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -60 \int e^{x^2-40 x+\frac {30}{x}} x^{x-1}dx-78 \int e^{x^2-40 x+\frac {30}{x}} x^{x+1}dx+4 \int e^{x^2-40 x+\frac {30}{x}} x^{x+2}dx-2 \int \frac {\int e^{x^2-40 x+\frac {30}{x}} x^{x+1}dx}{x}dx+2 \log (x) \int e^{x^2-40 x+\frac {30}{x}} x^{x+1}dx+2 \int e^{x^2-40 x+\frac {30}{x}} x^xdx-78 \int e^{\frac {2 \left (x^3-40 x^2+30\right )}{x}} x^{2 x}dx-60 \int e^{\frac {2 \left (x^3-40 x^2+30\right )}{x}} x^{2 x-2}dx+4 \int e^{\frac {2 \left (x^3-40 x^2+30\right )}{x}} x^{2 x+1}dx-2 \int \frac {\int e^{\frac {2 \left (x^3-40 x^2+30\right )}{x}} x^{2 x}dx}{x}dx+2 \log (x) \int e^{\frac {2 \left (x^3-40 x^2+30\right )}{x}} x^{2 x}dx+x^2-x\)

Input:

Int[(-x^2 + 2*x^3 + E^((2*(30 - 40*x^2 + x^3 + x^2*Log[x]))/x)*(-60 - 78*x 
^2 + 4*x^3 + 2*x^2*Log[x]) + E^((30 - 40*x^2 + x^3 + x^2*Log[x])/x)*(-60*x 
 + 2*x^2 - 78*x^3 + 4*x^4 + 2*x^3*Log[x]))/x^2,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76

method	result	size
risch	\(x^{2}+2 x^{x} {\mathrm e}^{\frac {x^{3}-40 x^{2}+30}{x}} x +x^{2 x} {\mathrm e}^{\frac {2 x^{3}-80 x^{2}+60}{x}}-x\)	\(51\)
default	\(-x +2 x \,{\mathrm e}^{\frac {x^{2} \ln \left (x \right )+x^{3}-40 x^{2}+30}{x}}+{\mathrm e}^{\frac {2 x^{2} \ln \left (x \right )+2 x^{3}-80 x^{2}+60}{x}}+x^{2}\)	\(54\)
parallelrisch	\(-x +2 x \,{\mathrm e}^{\frac {x^{2} \ln \left (x \right )+x^{3}-40 x^{2}+30}{x}}+{\mathrm e}^{\frac {2 x^{2} \ln \left (x \right )+2 x^{3}-80 x^{2}+60}{x}}+x^{2}\)	\(55\)

Input:

int(((2*x^2*ln(x)+4*x^3-78*x^2-60)*exp((x^2*ln(x)+x^3-40*x^2+30)/x)^2+(2*x 
^3*ln(x)+4*x^4-78*x^3+2*x^2-60*x)*exp((x^2*ln(x)+x^3-40*x^2+30)/x)+2*x^3-x 
^2)/x^2,x,method=_RETURNVERBOSE)
 

Output:

x^2+2*x^x*exp((x^3-40*x^2+30)/x)*x+(x^x)^2*exp(2*(x^3-40*x^2+30)/x)-x
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {-x^2+2 x^3+e^{\frac {2 \left (30-40 x^2+x^3+x^2 \log (x)\right )}{x}} \left (-60-78 x^2+4 x^3+2 x^2 \log (x)\right )+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} \left (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x)\right )}{x^2} \, dx=x^{2} + 2 \, x e^{\left (\frac {x^{3} + x^{2} \log \left (x\right ) - 40 \, x^{2} + 30}{x}\right )} - x + e^{\left (\frac {2 \, {\left (x^{3} + x^{2} \log \left (x\right ) - 40 \, x^{2} + 30\right )}}{x}\right )} \] Input:

integrate(((2*x^2*log(x)+4*x^3-78*x^2-60)*exp((x^2*log(x)+x^3-40*x^2+30)/x 
)^2+(2*x^3*log(x)+4*x^4-78*x^3+2*x^2-60*x)*exp((x^2*log(x)+x^3-40*x^2+30)/ 
x)+2*x^3-x^2)/x^2,x, algorithm="fricas")
 

Output:

x^2 + 2*x*e^((x^3 + x^2*log(x) - 40*x^2 + 30)/x) - x + e^(2*(x^3 + x^2*log 
(x) - 40*x^2 + 30)/x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).

Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {-x^2+2 x^3+e^{\frac {2 \left (30-40 x^2+x^3+x^2 \log (x)\right )}{x}} \left (-60-78 x^2+4 x^3+2 x^2 \log (x)\right )+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} \left (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x)\right )}{x^2} \, dx=x^{2} + 2 x e^{\frac {x^{3} + x^{2} \log {\left (x \right )} - 40 x^{2} + 30}{x}} - x + e^{\frac {2 \left (x^{3} + x^{2} \log {\left (x \right )} - 40 x^{2} + 30\right )}{x}} \] Input:

integrate(((2*x**2*ln(x)+4*x**3-78*x**2-60)*exp((x**2*ln(x)+x**3-40*x**2+3 
0)/x)**2+(2*x**3*ln(x)+4*x**4-78*x**3+2*x**2-60*x)*exp((x**2*ln(x)+x**3-40 
*x**2+30)/x)+2*x**3-x**2)/x**2,x)
 

Output:

x**2 + 2*x*exp((x**3 + x**2*log(x) - 40*x**2 + 30)/x) - x + exp(2*(x**3 + 
x**2*log(x) - 40*x**2 + 30)/x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {-x^2+2 x^3+e^{\frac {2 \left (30-40 x^2+x^3+x^2 \log (x)\right )}{x}} \left (-60-78 x^2+4 x^3+2 x^2 \log (x)\right )+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} \left (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x)\right )}{x^2} \, dx=x^{2} + {\left (2 \, x e^{\left (x^{2} + x \log \left (x\right ) + 40 \, x + \frac {30}{x}\right )} + e^{\left (2 \, x^{2} + 2 \, x \log \left (x\right ) + \frac {60}{x}\right )}\right )} e^{\left (-80 \, x\right )} - x \] Input:

integrate(((2*x^2*log(x)+4*x^3-78*x^2-60)*exp((x^2*log(x)+x^3-40*x^2+30)/x 
)^2+(2*x^3*log(x)+4*x^4-78*x^3+2*x^2-60*x)*exp((x^2*log(x)+x^3-40*x^2+30)/ 
x)+2*x^3-x^2)/x^2,x, algorithm="maxima")
 

Output:

x^2 + (2*x*e^(x^2 + x*log(x) + 40*x + 30/x) + e^(2*x^2 + 2*x*log(x) + 60/x 
))*e^(-80*x) - x
 

Giac [F]

\[ \int \frac {-x^2+2 x^3+e^{\frac {2 \left (30-40 x^2+x^3+x^2 \log (x)\right )}{x}} \left (-60-78 x^2+4 x^3+2 x^2 \log (x)\right )+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} \left (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x)\right )}{x^2} \, dx=\int { \frac {2 \, x^{3} - x^{2} + 2 \, {\left (2 \, x^{3} + x^{2} \log \left (x\right ) - 39 \, x^{2} - 30\right )} e^{\left (\frac {2 \, {\left (x^{3} + x^{2} \log \left (x\right ) - 40 \, x^{2} + 30\right )}}{x}\right )} + 2 \, {\left (2 \, x^{4} + x^{3} \log \left (x\right ) - 39 \, x^{3} + x^{2} - 30 \, x\right )} e^{\left (\frac {x^{3} + x^{2} \log \left (x\right ) - 40 \, x^{2} + 30}{x}\right )}}{x^{2}} \,d x } \] Input:

integrate(((2*x^2*log(x)+4*x^3-78*x^2-60)*exp((x^2*log(x)+x^3-40*x^2+30)/x 
)^2+(2*x^3*log(x)+4*x^4-78*x^3+2*x^2-60*x)*exp((x^2*log(x)+x^3-40*x^2+30)/ 
x)+2*x^3-x^2)/x^2,x, algorithm="giac")
 

Output:

integrate((2*x^3 - x^2 + 2*(2*x^3 + x^2*log(x) - 39*x^2 - 30)*e^(2*(x^3 + 
x^2*log(x) - 40*x^2 + 30)/x) + 2*(2*x^4 + x^3*log(x) - 39*x^3 + x^2 - 30*x 
)*e^((x^3 + x^2*log(x) - 40*x^2 + 30)/x))/x^2, x)
 

Mupad [B] (verification not implemented)

Time = 4.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {-x^2+2 x^3+e^{\frac {2 \left (30-40 x^2+x^3+x^2 \log (x)\right )}{x}} \left (-60-78 x^2+4 x^3+2 x^2 \log (x)\right )+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} \left (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x)\right )}{x^2} \, dx=x^{2\,x}\,{\mathrm {e}}^{\frac {60}{x}-80\,x+2\,x^2}-x+x^2+2\,x\,x^x\,{\mathrm {e}}^{\frac {30}{x}-40\,x+x^2} \] Input:

int((exp((2*(x^2*log(x) - 40*x^2 + x^3 + 30))/x)*(2*x^2*log(x) - 78*x^2 + 
4*x^3 - 60) + exp((x^2*log(x) - 40*x^2 + x^3 + 30)/x)*(2*x^3*log(x) - 60*x 
 + 2*x^2 - 78*x^3 + 4*x^4) - x^2 + 2*x^3)/x^2,x)
 

Output:

x^(2*x)*exp(60/x - 80*x + 2*x^2) - x + x^2 + 2*x*x^x*exp(30/x - 40*x + x^2 
)
 

Reduce [F]

\[ \int \frac {-x^2+2 x^3+e^{\frac {2 \left (30-40 x^2+x^3+x^2 \log (x)\right )}{x}} \left (-60-78 x^2+4 x^3+2 x^2 \log (x)\right )+e^{\frac {30-40 x^2+x^3+x^2 \log (x)}{x}} \left (-60 x+2 x^2-78 x^3+4 x^4+2 x^3 \log (x)\right )}{x^2} \, dx=\int \frac {\left (2 \,\mathrm {log}\left (x \right ) x^{2}+4 x^{3}-78 x^{2}-60\right ) \left ({\mathrm e}^{\frac {\mathrm {log}\left (x \right ) x^{2}+x^{3}-40 x^{2}+30}{x}}\right )^{2}+\left (2 \,\mathrm {log}\left (x \right ) x^{3}+4 x^{4}-78 x^{3}+2 x^{2}-60 x \right ) {\mathrm e}^{\frac {\mathrm {log}\left (x \right ) x^{2}+x^{3}-40 x^{2}+30}{x}}+2 x^{3}-x^{2}}{x^{2}}d x \] Input:

int(((2*x^2*log(x)+4*x^3-78*x^2-60)*exp((x^2*log(x)+x^3-40*x^2+30)/x)^2+(2 
*x^3*log(x)+4*x^4-78*x^3+2*x^2-60*x)*exp((x^2*log(x)+x^3-40*x^2+30)/x)+2*x 
^3-x^2)/x^2,x)
 

Output:

int(((2*x^2*log(x)+4*x^3-78*x^2-60)*exp((x^2*log(x)+x^3-40*x^2+30)/x)^2+(2 
*x^3*log(x)+4*x^4-78*x^3+2*x^2-60*x)*exp((x^2*log(x)+x^3-40*x^2+30)/x)+2*x 
^3-x^2)/x^2,x)