\(\int \frac {e^{\frac {4}{2+x}} (120+48 x-18 x^2-32 x^3-6 x^4)}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx\) [1832]

Optimal result

Integrand size = 70, antiderivative size = 28 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=\frac {2 e^{\frac {4}{2+x}}}{x \left (x^2-\frac {3 (4+x)}{x}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=-\frac {2 e^{\frac {4}{2+x}}}{12+3 x-x^3} \] Input:

Integrate[(E^(4/(2 + x))*(120 + 48*x - 18*x^2 - 32*x^3 - 6*x^4))/(576 + 86 
4*x + 468*x^2 + 12*x^3 - 111*x^4 - 48*x^5 - 2*x^6 + 4*x^7 + x^8),x]
 

Output:

(-2*E^(4/(2 + x)))/(12 + 3*x - x^3)
 

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.

\(\displaystyle \int \frac {e^{\frac {4}{x+2}} \left (-6 x^4-32 x^3-18 x^2+48 x+120\right )}{x^8+4 x^7-2 x^6-48 x^5-111 x^4+12 x^3+468 x^2+864 x+576} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {9 e^{\frac {4}{x+2}} \left (-6 x^4-32 x^3-18 x^2+48 x+120\right )}{1372 (x+2)}+\frac {e^{\frac {4}{x+2}} \left (-9 x^2+11 x+19\right ) \left (-6 x^4-32 x^3-18 x^2+48 x+120\right )}{1372 \left (x^3-3 x-12\right )}+\frac {e^{\frac {4}{x+2}} \left (-6 x^4-32 x^3-18 x^2+48 x+120\right )}{196 (x+2)^2}+\frac {e^{\frac {4}{x+2}} \left (9 x^2-4 x-47\right ) \left (-6 x^4-32 x^3-18 x^2+48 x+120\right )}{196 \left (x^3-3 x-12\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 6 \int \frac {e^{\frac {4}{x+2}}}{\left (x^3-3 x-12\right )^2}dx+\frac {94}{49} \int \frac {e^{\frac {4}{x+2}}}{x^3-3 x-12}dx+\frac {8}{49} \int \frac {e^{\frac {4}{x+2}} x}{x^3-3 x-12}dx-6 \int \frac {e^{\frac {4}{x+2}} x^2}{\left (x^3-3 x-12\right )^2}dx-\frac {18}{49} \int \frac {e^{\frac {4}{x+2}} x^2}{x^3-3 x-12}dx-\frac {18}{49} \operatorname {ExpIntegralEi}\left (\frac {4}{x+2}\right )-\frac {1}{7} e^{\frac {4}{x+2}}\)

Input:

Int[(E^(4/(2 + x))*(120 + 48*x - 18*x^2 - 32*x^3 - 6*x^4))/(576 + 864*x + 
468*x^2 + 12*x^3 - 111*x^4 - 48*x^5 - 2*x^6 + 4*x^7 + x^8),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75

method	result	size
gosper	\(\frac {2 \,{\mathrm e}^{\frac {4}{2+x}}}{x^{3}-3 x -12}\)	\(21\)
risch	\(\frac {2 \,{\mathrm e}^{\frac {4}{2+x}}}{x^{3}-3 x -12}\)	\(21\)
parallelrisch	\(\frac {2 \,{\mathrm e}^{\frac {4}{2+x}}}{x^{3}-3 x -12}\)	\(21\)
norman	\(\frac {2 x \,{\mathrm e}^{\frac {4}{2+x}}+4 \,{\mathrm e}^{\frac {4}{2+x}}}{x^{4}+2 x^{3}-3 x^{2}-18 x -24}\)	\(44\)
orering	\(-\frac {\left (2+x \right )^{2} \left (x^{3}-3 x -12\right ) \left (-6 x^{4}-32 x^{3}-18 x^{2}+48 x +120\right ) {\mathrm e}^{\frac {4}{2+x}}}{\left (3 x^{4}+16 x^{3}+9 x^{2}-24 x -60\right ) \left (x^{8}+4 x^{7}-2 x^{6}-48 x^{5}-111 x^{4}+12 x^{3}+468 x^{2}+864 x +576\right )}\)	\(106\)
derivativedivides	\(\frac {4 \,{\mathrm e}^{\frac {4}{2+x}} \left (\frac {112}{\left (2+x \right )^{2}}+\frac {40}{2+x}-32\right )}{35 \left (\frac {448}{\left (2+x \right )^{3}}-\frac {288}{\left (2+x \right )^{2}}+\frac {192}{2+x}-32\right )}+\frac {4 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (7 \textit {\_Z}^{3}-18 \textit {\_Z}^{2}+48 \textit {\_Z} -32\right )}{\sum }\frac {\left (7 \textit {\_R1}^{2}+3 \textit {\_R1} -70\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{105}-\frac {32 \,{\mathrm e}^{\frac {4}{2+x}} \left (\frac {112}{\left (2+x \right )^{2}}-\frac {80}{2+x}+8\right )}{105 \left (\frac {448}{\left (2+x \right )^{3}}-\frac {288}{\left (2+x \right )^{2}}+\frac {192}{2+x}-32\right )}-\frac {32 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (7 \textit {\_Z}^{3}-18 \textit {\_Z}^{2}+48 \textit {\_Z} -32\right )}{\sum }\frac {\textit {\_R1} \left (7 \textit {\_R1} -27\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{315}+\frac {2 \,{\mathrm e}^{\frac {4}{2+x}} \left (\frac {16}{\left (2+x \right )^{2}}+\frac {80}{2+x}-16\right )}{7 \left (\frac {448}{\left (2+x \right )^{3}}-\frac {288}{\left (2+x \right )^{2}}+\frac {192}{2+x}-32\right )}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (7 \textit {\_Z}^{3}-18 \textit {\_Z}^{2}+48 \textit {\_Z} -32\right )}{\sum }\frac {\left (\textit {\_R1}^{2}+34 \textit {\_R1} -20\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{21}-\frac {12 \,{\mathrm e}^{\frac {4}{2+x}} \left (\frac {1264}{\left (2+x \right )^{2}}-\frac {320}{2+x}+16\right )}{245 \left (\frac {448}{\left (2+x \right )^{3}}-\frac {288}{\left (2+x \right )^{2}}+\frac {192}{2+x}-32\right )}-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (7 \textit {\_Z}^{3}-18 \textit {\_Z}^{2}+48 \textit {\_Z} -32\right )}{\sum }\frac {\left (263 \textit {\_R1}^{2}-48 \textit {\_R1} -80\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{245}-\frac {{\mathrm e}^{\frac {4}{2+x}}}{7}+\frac {4 \,{\mathrm e}^{\frac {4}{2+x}} \left (\frac {6896}{\left (2+x \right )^{2}}-\frac {7360}{2+x}+1264\right )}{735 \left (\frac {448}{\left (2+x \right )^{3}}-\frac {288}{\left (2+x \right )^{2}}+\frac {192}{2+x}-32\right )}+\frac {32 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (7 \textit {\_Z}^{3}-18 \textit {\_Z}^{2}+48 \textit {\_Z} -32\right )}{\sum }\frac {\left (172 \textit {\_R1}^{2}-447 \textit {\_R1} +270\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{2205}\)	\(546\)
default	\(\frac {4 \,{\mathrm e}^{\frac {4}{2+x}} \left (\frac {112}{\left (2+x \right )^{2}}+\frac {40}{2+x}-32\right )}{35 \left (\frac {448}{\left (2+x \right )^{3}}-\frac {288}{\left (2+x \right )^{2}}+\frac {192}{2+x}-32\right )}+\frac {4 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (7 \textit {\_Z}^{3}-18 \textit {\_Z}^{2}+48 \textit {\_Z} -32\right )}{\sum }\frac {\left (7 \textit {\_R1}^{2}+3 \textit {\_R1} -70\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{105}-\frac {32 \,{\mathrm e}^{\frac {4}{2+x}} \left (\frac {112}{\left (2+x \right )^{2}}-\frac {80}{2+x}+8\right )}{105 \left (\frac {448}{\left (2+x \right )^{3}}-\frac {288}{\left (2+x \right )^{2}}+\frac {192}{2+x}-32\right )}-\frac {32 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (7 \textit {\_Z}^{3}-18 \textit {\_Z}^{2}+48 \textit {\_Z} -32\right )}{\sum }\frac {\textit {\_R1} \left (7 \textit {\_R1} -27\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{315}+\frac {2 \,{\mathrm e}^{\frac {4}{2+x}} \left (\frac {16}{\left (2+x \right )^{2}}+\frac {80}{2+x}-16\right )}{7 \left (\frac {448}{\left (2+x \right )^{3}}-\frac {288}{\left (2+x \right )^{2}}+\frac {192}{2+x}-32\right )}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (7 \textit {\_Z}^{3}-18 \textit {\_Z}^{2}+48 \textit {\_Z} -32\right )}{\sum }\frac {\left (\textit {\_R1}^{2}+34 \textit {\_R1} -20\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{21}-\frac {12 \,{\mathrm e}^{\frac {4}{2+x}} \left (\frac {1264}{\left (2+x \right )^{2}}-\frac {320}{2+x}+16\right )}{245 \left (\frac {448}{\left (2+x \right )^{3}}-\frac {288}{\left (2+x \right )^{2}}+\frac {192}{2+x}-32\right )}-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (7 \textit {\_Z}^{3}-18 \textit {\_Z}^{2}+48 \textit {\_Z} -32\right )}{\sum }\frac {\left (263 \textit {\_R1}^{2}-48 \textit {\_R1} -80\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{245}-\frac {{\mathrm e}^{\frac {4}{2+x}}}{7}+\frac {4 \,{\mathrm e}^{\frac {4}{2+x}} \left (\frac {6896}{\left (2+x \right )^{2}}-\frac {7360}{2+x}+1264\right )}{735 \left (\frac {448}{\left (2+x \right )^{3}}-\frac {288}{\left (2+x \right )^{2}}+\frac {192}{2+x}-32\right )}+\frac {32 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (7 \textit {\_Z}^{3}-18 \textit {\_Z}^{2}+48 \textit {\_Z} -32\right )}{\sum }\frac {\left (172 \textit {\_R1}^{2}-447 \textit {\_R1} +270\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{2205}\)	\(546\)

Input:

int((-6*x^4-32*x^3-18*x^2+48*x+120)*exp(4/(2+x))/(x^8+4*x^7-2*x^6-48*x^5-1 
11*x^4+12*x^3+468*x^2+864*x+576),x,method=_RETURNVERBOSE)
 

Output:

2*exp(4/(2+x))/(x^3-3*x-12)
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=\frac {2 \, e^{\left (\frac {4}{x + 2}\right )}}{x^{3} - 3 \, x - 12} \] Input:

integrate((-6*x^4-32*x^3-18*x^2+48*x+120)*exp(4/(2+x))/(x^8+4*x^7-2*x^6-48 
*x^5-111*x^4+12*x^3+468*x^2+864*x+576),x, algorithm="fricas")
 

Output:

2*e^(4/(x + 2))/(x^3 - 3*x - 12)
 

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=\frac {2 e^{\frac {4}{x + 2}}}{x^{3} - 3 x - 12} \] Input:

integrate((-6*x**4-32*x**3-18*x**2+48*x+120)*exp(4/(2+x))/(x**8+4*x**7-2*x 
**6-48*x**5-111*x**4+12*x**3+468*x**2+864*x+576),x)
 

Output:

2*exp(4/(x + 2))/(x**3 - 3*x - 12)
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=\frac {2 \, e^{\left (\frac {4}{x + 2}\right )}}{x^{3} - 3 \, x - 12} \] Input:

integrate((-6*x^4-32*x^3-18*x^2+48*x+120)*exp(4/(2+x))/(x^8+4*x^7-2*x^6-48 
*x^5-111*x^4+12*x^3+468*x^2+864*x+576),x, algorithm="maxima")
 

Output:

2*e^(4/(x + 2))/(x^3 - 3*x - 12)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=-\frac {2 \, e^{\left (\frac {4}{x + 2}\right )}}{{\left (x + 2\right )}^{3} {\left (\frac {6}{x + 2} - \frac {9}{{\left (x + 2\right )}^{2}} + \frac {14}{{\left (x + 2\right )}^{3}} - 1\right )}} \] Input:

integrate((-6*x^4-32*x^3-18*x^2+48*x+120)*exp(4/(2+x))/(x^8+4*x^7-2*x^6-48 
*x^5-111*x^4+12*x^3+468*x^2+864*x+576),x, algorithm="giac")
 

Output:

-2*e^(4/(x + 2))/((x + 2)^3*(6/(x + 2) - 9/(x + 2)^2 + 14/(x + 2)^3 - 1))
 

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=-\frac {2\,{\mathrm {e}}^{\frac {4}{x+2}}}{-x^3+3\,x+12} \] Input:

int(-(exp(4/(x + 2))*(18*x^2 - 48*x + 32*x^3 + 6*x^4 - 120))/(864*x + 468* 
x^2 + 12*x^3 - 111*x^4 - 48*x^5 - 2*x^6 + 4*x^7 + x^8 + 576),x)
 

Output:

-(2*exp(4/(x + 2)))/(3*x - x^3 + 12)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=\frac {2 e^{\frac {4}{x +2}}}{x^{3}-3 x -12} \] Input:

int((-6*x^4-32*x^3-18*x^2+48*x+120)*exp(4/(2+x))/(x^8+4*x^7-2*x^6-48*x^5-1 
11*x^4+12*x^3+468*x^2+864*x+576),x)
 

Output:

(2*e**(4/(x + 2)))/(x**3 - 3*x - 12)