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

Optimal result

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

[Out]

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

Rubi [F]

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

[In]

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]

[Out]

-1/7*E^(4/(2 + x)) - (18*ExpIntegralEi[4/(2 + x)])/49 + 6*Defer[Int][E^(4/(2 + x))/(-12 - 3*x + x^3)^2, x] - 6
*Defer[Int][(E^(4/(2 + x))*x^2)/(-12 - 3*x + x^3)^2, x] + (94*Defer[Int][E^(4/(2 + x))/(-12 - 3*x + x^3), x])/
49 + (8*Defer[Int][(E^(4/(2 + x))*x)/(-12 - 3*x + x^3), x])/49 - (18*Defer[Int][(E^(4/(2 + x))*x^2)/(-12 - 3*x
+ x^3), x])/49

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^{\frac {4}{2+x}} \left (60+24 x-9 x^2-16 x^3-3 x^4\right )}{\left (24+18 x+3 x^2-2 x^3-x^4\right )^2} \, dx \\ & = 2 \int \frac {e^{\frac {4}{2+x}} \left (60+24 x-9 x^2-16 x^3-3 x^4\right )}{\left (24+18 x+3 x^2-2 x^3-x^4\right )^2} \, dx \\ & = 2 \int \left (\frac {2 e^{\frac {4}{2+x}}}{7 (2+x)^2}+\frac {9 e^{\frac {4}{2+x}}}{49 (2+x)}-\frac {3 e^{\frac {4}{2+x}} \left (-1+x^2\right )}{\left (-12-3 x+x^3\right )^2}+\frac {e^{\frac {4}{2+x}} \left (47+4 x-9 x^2\right )}{49 \left (-12-3 x+x^3\right )}\right ) \, dx \\ & = \frac {2}{49} \int \frac {e^{\frac {4}{2+x}} \left (47+4 x-9 x^2\right )}{-12-3 x+x^3} \, dx+\frac {18}{49} \int \frac {e^{\frac {4}{2+x}}}{2+x} \, dx+\frac {4}{7} \int \frac {e^{\frac {4}{2+x}}}{(2+x)^2} \, dx-6 \int \frac {e^{\frac {4}{2+x}} \left (-1+x^2\right )}{\left (-12-3 x+x^3\right )^2} \, dx \\ & = -\frac {1}{7} e^{\frac {4}{2+x}}-\frac {18 \operatorname {ExpIntegralEi}\left (\frac {4}{2+x}\right )}{49}+\frac {2}{49} \int \left (\frac {47 e^{\frac {4}{2+x}}}{-12-3 x+x^3}+\frac {4 e^{\frac {4}{2+x}} x}{-12-3 x+x^3}-\frac {9 e^{\frac {4}{2+x}} x^2}{-12-3 x+x^3}\right ) \, dx-6 \int \left (-\frac {e^{\frac {4}{2+x}}}{\left (-12-3 x+x^3\right )^2}+\frac {e^{\frac {4}{2+x}} x^2}{\left (-12-3 x+x^3\right )^2}\right ) \, dx \\ & = -\frac {1}{7} e^{\frac {4}{2+x}}-\frac {18 \operatorname {ExpIntegralEi}\left (\frac {4}{2+x}\right )}{49}+\frac {8}{49} \int \frac {e^{\frac {4}{2+x}} x}{-12-3 x+x^3} \, dx-\frac {18}{49} \int \frac {e^{\frac {4}{2+x}} x^2}{-12-3 x+x^3} \, dx+\frac {94}{49} \int \frac {e^{\frac {4}{2+x}}}{-12-3 x+x^3} \, dx+6 \int \frac {e^{\frac {4}{2+x}}}{\left (-12-3 x+x^3\right )^2} \, dx-6 \int \frac {e^{\frac {4}{2+x}} x^2}{\left (-12-3 x+x^3\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=-\frac {2 e^{\frac {4}{2+x}}}{12+3 x-x^3} \]

[In]

Integrate[(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 - 4
8*x^5 - 2*x^6 + 4*x^7 + x^8),x]

[Out]

(-2*E^(4/(2 + x)))/(12 + 3*x - x^3)

Maple [A] (verified)

Time = 0.27 (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	\(\frac {2 x \,{\mathrm e}^{\frac {4}{2+x}}+4 \,{\mathrm e}^{\frac {4}{2+x}}}{x^{4}+2 x^{3}-3 x^{2}-18 x -24}\)	\(44\)
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 {Ei}_{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 {Ei}_{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 {Ei}_{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 {Ei}_{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 {Ei}_{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 {Ei}_{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 {Ei}_{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 {Ei}_{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 {Ei}_{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 {Ei}_{1}\left (-\frac {4}{2+x}+\textit {\_R1} \right )}{7 \textit {\_R1}^{2}-12 \textit {\_R1} +16}\right )}{2205}\)	\(546\)

[In]

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^5-111*x^4+12*x^3+468*x^2+864*x+576),x,m
ethod=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=\frac {2 \, e^{\left (\frac {4}{x + 2}\right )}}{x^{3} - 3 \, x - 12} \]

[In]

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+57
6),x, algorithm="fricas")

[Out]

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

[In]

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)

[Out]

2*exp(4/(x + 2))/(x**3 - 3*x - 12)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=\frac {2 \, e^{\left (\frac {4}{x + 2}\right )}}{x^{3} - 3 \, x - 12} \]

[In]

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+57
6),x, algorithm="maxima")

[Out]

2*e^(4/(x + 2))/(x^3 - 3*x - 12)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=-\frac {2 \, e^{\left (\frac {4}{x + 2}\right )}}{{\left (x + 2\right )}^{3} {\left (\frac {6}{x + 2} - \frac {9}{{\left (x + 2\right )}^{2}} + \frac {14}{{\left (x + 2\right )}^{3}} - 1\right )}} \]

[In]

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+57
6),x, algorithm="giac")

[Out]

-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 = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {4}{2+x}} \left (120+48 x-18 x^2-32 x^3-6 x^4\right )}{576+864 x+468 x^2+12 x^3-111 x^4-48 x^5-2 x^6+4 x^7+x^8} \, dx=-\frac {2\,{\mathrm {e}}^{\frac {4}{x+2}}}{-x^3+3\,x+12} \]

[In]

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)

[Out]

-(2*exp(4/(x + 2)))/(3*x - x^3 + 12)