\(\int \frac {e^{\frac {2 x^2-x^6+x^2 \log (x)}{2+5 x+2 x^2}} (10 x+15 x^2+2 x^3-12 x^5-25 x^6-8 x^7+(4 x+5 x^2) \log (x))}{4+20 x+33 x^2+20 x^3+4 x^4} \, dx\) [24]

Optimal result

Integrand size = 96, antiderivative size = 27 \[ \int \frac {e^{\frac {2 x^2-x^6+x^2 \log (x)}{2+5 x+2 x^2}} \left (10 x+15 x^2+2 x^3-12 x^5-25 x^6-8 x^7+\left (4 x+5 x^2\right ) \log (x)\right )}{4+20 x+33 x^2+20 x^3+4 x^4} \, dx=e^{\frac {x \left (2-x^4+\log (x)\right )}{4+2 x+\frac {2+x}{x}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {2 x^2-x^6+x^2 \log (x)}{2+5 x+2 x^2}} \left (10 x+15 x^2+2 x^3-12 x^5-25 x^6-8 x^7+\left (4 x+5 x^2\right ) \log (x)\right )}{4+20 x+33 x^2+20 x^3+4 x^4} \, dx=e^{-\frac {x^2 \left (-2+x^4\right )}{2+5 x+2 x^2}} x^{\frac {x^2}{2+5 x+2 x^2}} \] Input:

Integrate[(E^((2*x^2 - x^6 + x^2*Log[x])/(2 + 5*x + 2*x^2))*(10*x + 15*x^2 
 + 2*x^3 - 12*x^5 - 25*x^6 - 8*x^7 + (4*x + 5*x^2)*Log[x]))/(4 + 20*x + 33 
*x^2 + 20*x^3 + 4*x^4),x]
 

Output:

x^(x^2/(2 + 5*x + 2*x^2))/E^((x^2*(-2 + x^4))/(2 + 5*x + 2*x^2))
 

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.

\(\displaystyle \int \frac {e^{\frac {-x^6+2 x^2+x^2 \log (x)}{2 x^2+5 x+2}} \left (-8 x^7-25 x^6-12 x^5+2 x^3+15 x^2+\left (5 x^2+4 x\right ) \log (x)+10 x\right )}{4 x^4+20 x^3+33 x^2+20 x+4} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {4 e^{\frac {-x^6+2 x^2+x^2 \log (x)}{2 x^2+5 x+2}} \left (-8 x^7-25 x^6-12 x^5+2 x^3+15 x^2+\left (5 x^2+4 x\right ) \log (x)+10 x\right )}{27 (x+2)}-\frac {8 e^{\frac {-x^6+2 x^2+x^2 \log (x)}{2 x^2+5 x+2}} \left (-8 x^7-25 x^6-12 x^5+2 x^3+15 x^2+\left (5 x^2+4 x\right ) \log (x)+10 x\right )}{27 (2 x+1)}+\frac {e^{\frac {-x^6+2 x^2+x^2 \log (x)}{2 x^2+5 x+2}} \left (-8 x^7-25 x^6-12 x^5+2 x^3+15 x^2+\left (5 x^2+4 x\right ) \log (x)+10 x\right )}{9 (x+2)^2}+\frac {4 e^{\frac {-x^6+2 x^2+x^2 \log (x)}{2 x^2+5 x+2}} \left (-8 x^7-25 x^6-12 x^5+2 x^3+15 x^2+\left (5 x^2+4 x\right ) \log (x)+10 x\right )}{9 (2 x+1)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1} \left (-8 x^6-25 x^5-12 x^4+2 x^2+15 x+(5 x+4) \log (x)+10\right )}{(x+2)^2 (2 x+1)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {10 e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{(x+2)^2 (2 x+1)^2}+\frac {15 e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+2}}{(x+2)^2 (2 x+1)^2}+\frac {2 e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+3}}{(x+2)^2 (2 x+1)^2}-\frac {12 e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+5}}{(x+2)^2 (2 x+1)^2}-\frac {25 e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+6}}{(x+2)^2 (2 x+1)^2}-\frac {8 e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+7}}{(x+2)^2 (2 x+1)^2}+\frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} (5 x+4) x^{\frac {x^2}{2 x^2+5 x+2}+1} \log (x)}{(x+2)^2 (2 x+1)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2}{3} \log (x) \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{(x+2)^2}dx+\frac {10}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{(x+2)^2}dx+\frac {5}{3} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+2}}{(x+2)^2}dx+\frac {2}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+3}}{(x+2)^2}dx-\frac {4}{3} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+5}}{(x+2)^2}dx-\frac {25}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+6}}{(x+2)^2}dx-\frac {8}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+7}}{(x+2)^2}dx-\frac {1}{3} \log (x) \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{x+2}dx+\frac {40}{27} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{x+2}dx+\frac {20}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+2}}{x+2}dx+\frac {8}{27} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+3}}{x+2}dx-\frac {16}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+5}}{x+2}dx-\frac {100}{27} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+6}}{x+2}dx-\frac {32}{27} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+7}}{x+2}dx+\frac {2}{3} \log (x) \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{(2 x+1)^2}dx+\frac {40}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{(2 x+1)^2}dx+\frac {20}{3} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+2}}{(2 x+1)^2}dx+\frac {8}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+3}}{(2 x+1)^2}dx-\frac {16}{3} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+5}}{(2 x+1)^2}dx-\frac {100}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+6}}{(2 x+1)^2}dx-\frac {32}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+7}}{(2 x+1)^2}dx+\frac {2}{3} \log (x) \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{2 x+1}dx-\frac {80}{27} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{2 x+1}dx-\frac {40}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+2}}{2 x+1}dx-\frac {16}{27} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+3}}{2 x+1}dx+\frac {32}{9} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+5}}{2 x+1}dx+\frac {200}{27} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+6}}{2 x+1}dx+\frac {64}{27} \int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+7}}{2 x+1}dx+\frac {2}{3} \int \frac {\int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{(x+2)^2}dx}{x}dx+\frac {1}{3} \int \frac {\int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{x+2}dx}{x}dx-\frac {2}{3} \int \frac {\int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{(2 x+1)^2}dx}{x}dx-\frac {2}{3} \int \frac {\int \frac {e^{-\frac {x^2 \left (x^4-2\right )}{2 x^2+5 x+2}} x^{\frac {x^2}{2 x^2+5 x+2}+1}}{2 x+1}dx}{x}dx\)

Input:

Int[(E^((2*x^2 - x^6 + x^2*Log[x])/(2 + 5*x + 2*x^2))*(10*x + 15*x^2 + 2*x 
^3 - 12*x^5 - 25*x^6 - 8*x^7 + (4*x + 5*x^2)*Log[x]))/(4 + 20*x + 33*x^2 + 
 20*x^3 + 4*x^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 7.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

method	result	size
risch	\({\mathrm e}^{\frac {x^{2} \left (2-x^{4}+\ln \left (x \right )\right )}{\left (2+x \right ) \left (1+2 x \right )}}\)	\(27\)
parallelrisch	\({\mathrm e}^{\frac {x^{2} \left (2-x^{4}+\ln \left (x \right )\right )}{2 x^{2}+5 x +2}}\)	\(27\)

Input:

int(((5*x^2+4*x)*ln(x)-8*x^7-25*x^6-12*x^5+2*x^3+15*x^2+10*x)*exp((x^2*ln( 
x)-x^6+2*x^2)/(2*x^2+5*x+2))/(4*x^4+20*x^3+33*x^2+20*x+4),x,method=_RETURN 
VERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {2 x^2-x^6+x^2 \log (x)}{2+5 x+2 x^2}} \left (10 x+15 x^2+2 x^3-12 x^5-25 x^6-8 x^7+\left (4 x+5 x^2\right ) \log (x)\right )}{4+20 x+33 x^2+20 x^3+4 x^4} \, dx=e^{\left (-\frac {x^{6} - x^{2} \log \left (x\right ) - 2 \, x^{2}}{2 \, x^{2} + 5 \, x + 2}\right )} \] Input:

integrate(((5*x^2+4*x)*log(x)-8*x^7-25*x^6-12*x^5+2*x^3+15*x^2+10*x)*exp(( 
x^2*log(x)-x^6+2*x^2)/(2*x^2+5*x+2))/(4*x^4+20*x^3+33*x^2+20*x+4),x, algor 
ithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {2 x^2-x^6+x^2 \log (x)}{2+5 x+2 x^2}} \left (10 x+15 x^2+2 x^3-12 x^5-25 x^6-8 x^7+\left (4 x+5 x^2\right ) \log (x)\right )}{4+20 x+33 x^2+20 x^3+4 x^4} \, dx=e^{\frac {- x^{6} + x^{2} \log {\left (x \right )} + 2 x^{2}}{2 x^{2} + 5 x + 2}} \] Input:

integrate(((5*x**2+4*x)*ln(x)-8*x**7-25*x**6-12*x**5+2*x**3+15*x**2+10*x)* 
exp((x**2*ln(x)-x**6+2*x**2)/(2*x**2+5*x+2))/(4*x**4+20*x**3+33*x**2+20*x+ 
4),x)
 

Output:

exp((-x**6 + x**2*log(x) + 2*x**2)/(2*x**2 + 5*x + 2))
 

Maxima [F]

\[ \int \frac {e^{\frac {2 x^2-x^6+x^2 \log (x)}{2+5 x+2 x^2}} \left (10 x+15 x^2+2 x^3-12 x^5-25 x^6-8 x^7+\left (4 x+5 x^2\right ) \log (x)\right )}{4+20 x+33 x^2+20 x^3+4 x^4} \, dx=\int { -\frac {{\left (8 \, x^{7} + 25 \, x^{6} + 12 \, x^{5} - 2 \, x^{3} - 15 \, x^{2} - {\left (5 \, x^{2} + 4 \, x\right )} \log \left (x\right ) - 10 \, x\right )} e^{\left (-\frac {x^{6} - x^{2} \log \left (x\right ) - 2 \, x^{2}}{2 \, x^{2} + 5 \, x + 2}\right )}}{4 \, x^{4} + 20 \, x^{3} + 33 \, x^{2} + 20 \, x + 4} \,d x } \] Input:

integrate(((5*x^2+4*x)*log(x)-8*x^7-25*x^6-12*x^5+2*x^3+15*x^2+10*x)*exp(( 
x^2*log(x)-x^6+2*x^2)/(2*x^2+5*x+2))/(4*x^4+20*x^3+33*x^2+20*x+4),x, algor 
ithm="maxima")
 

Output:

-integrate((8*x^7 + 25*x^6 + 12*x^5 - 2*x^3 - 15*x^2 - (5*x^2 + 4*x)*log(x 
) - 10*x)*e^(-(x^6 - x^2*log(x) - 2*x^2)/(2*x^2 + 5*x + 2))/(4*x^4 + 20*x^ 
3 + 33*x^2 + 20*x + 4), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {e^{\frac {2 x^2-x^6+x^2 \log (x)}{2+5 x+2 x^2}} \left (10 x+15 x^2+2 x^3-12 x^5-25 x^6-8 x^7+\left (4 x+5 x^2\right ) \log (x)\right )}{4+20 x+33 x^2+20 x^3+4 x^4} \, dx=e^{\left (-\frac {x^{6}}{2 \, x^{2} + 5 \, x + 2} + \frac {x^{2} \log \left (x\right )}{2 \, x^{2} + 5 \, x + 2} + \frac {2 \, x^{2}}{2 \, x^{2} + 5 \, x + 2}\right )} \] Input:

integrate(((5*x^2+4*x)*log(x)-8*x^7-25*x^6-12*x^5+2*x^3+15*x^2+10*x)*exp(( 
x^2*log(x)-x^6+2*x^2)/(2*x^2+5*x+2))/(4*x^4+20*x^3+33*x^2+20*x+4),x, algor 
ithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {e^{\frac {2 x^2-x^6+x^2 \log (x)}{2+5 x+2 x^2}} \left (10 x+15 x^2+2 x^3-12 x^5-25 x^6-8 x^7+\left (4 x+5 x^2\right ) \log (x)\right )}{4+20 x+33 x^2+20 x^3+4 x^4} \, dx=x^{\frac {x^2}{2\,x^2+5\,x+2}}\,{\mathrm {e}}^{\frac {2\,x^2-x^6}{2\,x^2+5\,x+2}} \] Input:

int((exp((x^2*log(x) + 2*x^2 - x^6)/(5*x + 2*x^2 + 2))*(10*x + log(x)*(4*x 
 + 5*x^2) + 15*x^2 + 2*x^3 - 12*x^5 - 25*x^6 - 8*x^7))/(20*x + 33*x^2 + 20 
*x^3 + 4*x^4 + 4),x)
 

Output:

x^(x^2/(5*x + 2*x^2 + 2))*exp((2*x^2 - x^6)/(5*x + 2*x^2 + 2))
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {e^{\frac {2 x^2-x^6+x^2 \log (x)}{2+5 x+2 x^2}} \left (10 x+15 x^2+2 x^3-12 x^5-25 x^6-8 x^7+\left (4 x+5 x^2\right ) \log (x)\right )}{4+20 x+33 x^2+20 x^3+4 x^4} \, dx=\frac {e^{\frac {\mathrm {log}\left (x \right ) x^{2}}{2 x^{2}+5 x +2}} e}{e^{\frac {x^{6}+5 x +2}{2 x^{2}+5 x +2}}} \] Input:

int(((5*x^2+4*x)*log(x)-8*x^7-25*x^6-12*x^5+2*x^3+15*x^2+10*x)*exp((x^2*lo 
g(x)-x^6+2*x^2)/(2*x^2+5*x+2))/(4*x^4+20*x^3+33*x^2+20*x+4),x)
 

Output:

(e**((log(x)*x**2)/(2*x**2 + 5*x + 2))*e)/e**((x**6 + 5*x + 2)/(2*x**2 + 5 
*x + 2))