\(\int \frac {e^{4-x} (-6+6 x-2 x^2-x^3+x^4)}{180+60 x-175 x^2+30 x^3+55 x^4-30 x^5+5 x^6} \, dx\) [524]

Optimal result

Integrand size = 58, antiderivative size = 27 \[ \int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{180+60 x-175 x^2+30 x^3+55 x^4-30 x^5+5 x^6} \, dx=-\frac {e^{4-x} x}{5 \left (6+x-(3-x) x^2\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{180+60 x-175 x^2+30 x^3+55 x^4-30 x^5+5 x^6} \, dx=\int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{180+60 x-175 x^2+30 x^3+55 x^4-30 x^5+5 x^6} \, dx \]

[In]

Int[(E^(4 - x)*(-6 + 6*x - 2*x^2 - x^3 + x^4))/(180 + 60*x - 175*x^2 + 30*x^3 + 55*x^4 - 30*x^5 + 5*x^6),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{5 \left (6+x-3 x^2+x^3\right )^2} \, dx \\ & = \frac {1}{5} \int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{\left (6+x-3 x^2+x^3\right )^2} \, dx \\ & = \frac {1}{5} \int \left (\frac {e^{4-x} \left (-18-2 x+3 x^2\right )}{\left (6+x-3 x^2+x^3\right )^2}+\frac {e^{4-x} (2+x)}{6+x-3 x^2+x^3}\right ) \, dx \\ & = \frac {1}{5} \int \frac {e^{4-x} \left (-18-2 x+3 x^2\right )}{\left (6+x-3 x^2+x^3\right )^2} \, dx+\frac {1}{5} \int \frac {e^{4-x} (2+x)}{6+x-3 x^2+x^3} \, dx \\ & = \frac {1}{5} \int \left (-\frac {18 e^{4-x}}{\left (6+x-3 x^2+x^3\right )^2}-\frac {2 e^{4-x} x}{\left (6+x-3 x^2+x^3\right )^2}+\frac {3 e^{4-x} x^2}{\left (6+x-3 x^2+x^3\right )^2}\right ) \, dx+\frac {1}{5} \int \left (\frac {2 e^{4-x}}{6+x-3 x^2+x^3}+\frac {e^{4-x} x}{6+x-3 x^2+x^3}\right ) \, dx \\ & = \frac {1}{5} \int \frac {e^{4-x} x}{6+x-3 x^2+x^3} \, dx-\frac {2}{5} \int \frac {e^{4-x} x}{\left (6+x-3 x^2+x^3\right )^2} \, dx+\frac {2}{5} \int \frac {e^{4-x}}{6+x-3 x^2+x^3} \, dx+\frac {3}{5} \int \frac {e^{4-x} x^2}{\left (6+x-3 x^2+x^3\right )^2} \, dx-\frac {18}{5} \int \frac {e^{4-x}}{\left (6+x-3 x^2+x^3\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{180+60 x-175 x^2+30 x^3+55 x^4-30 x^5+5 x^6} \, dx=-\frac {e^{4-x} x}{5 \left (6+x-3 x^2+x^3\right )} \]

[In]

Integrate[(E^(4 - x)*(-6 + 6*x - 2*x^2 - x^3 + x^4))/(180 + 60*x - 175*x^2 + 30*x^3 + 55*x^4 - 30*x^5 + 5*x^6)
,x]

[Out]

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {x \,{\mathrm e}^{-x +4}}{5 \left (x^{3}-3 x^{2}+x +6\right )}\)	\(23\)
gosper	\(-\frac {{\mathrm e}^{4} x \,{\mathrm e}^{-x}}{5 \left (x^{3}-3 x^{2}+x +6\right )}\)	\(25\)
norman	\(-\frac {{\mathrm e}^{4} x \,{\mathrm e}^{-x}}{5 \left (x^{3}-3 x^{2}+x +6\right )}\)	\(25\)
parallelrisch	\(-\frac {{\mathrm e}^{4} x \,{\mathrm e}^{-x}}{5 \left (x^{3}-3 x^{2}+x +6\right )}\)	\(25\)
default	\(\frac {{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-x} \left (151 x^{2}-861 x -1206\right )}{643 x^{3}-1929 x^{2}+643 x +3858}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+\textit {\_Z} +6\right )}{\sum }\frac {\left (151 \textit {\_R1}^{2}-67 \textit {\_R1} -546\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (x -\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-6 \textit {\_R1} +1}\right )}{643}-\frac {6 \,{\mathrm e}^{-x} \left (12 x^{2}+21 x -49\right )}{643 \left (x^{3}-3 x^{2}+x +6\right )}+\frac {6 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+\textit {\_Z} +6\right )}{\sum }\frac {\left (12 \textit {\_R1}^{2}+33 \textit {\_R1} +29\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (x -\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-6 \textit {\_R1} +1}\right )}{643}+\frac {6 \,{\mathrm e}^{-x} \left (57 x^{2}-61 x -72\right )}{643 \left (x^{3}-3 x^{2}+x +6\right )}-\frac {6 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+\textit {\_Z} +6\right )}{\sum }\frac {\left (57 \textit {\_R1}^{2}-4 \textit {\_R1} -23\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (x -\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-6 \textit {\_R1} +1}\right )}{643}-\frac {2 \,{\mathrm e}^{-x} \left (110 x^{2}-129 x -342\right )}{643 \left (x^{3}-3 x^{2}+x +6\right )}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+\textit {\_Z} +6\right )}{\sum }\frac {\left (110 \textit {\_R1}^{2}-19 \textit {\_R1} -270\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (x -\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-6 \textit {\_R1} +1}\right )}{643}-\frac {{\mathrm e}^{-x} \left (201 x^{2}-452 x -660\right )}{643 \left (x^{3}-3 x^{2}+x +6\right )}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+\textit {\_Z} +6\right )}{\sum }\frac {\left (201 \textit {\_R1}^{2}-251 \textit {\_R1} -318\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (x -\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-6 \textit {\_R1} +1}\right )}{643}\right )}{5}\)	\(408\)

[In]

int((x^4-x^3-2*x^2+6*x-6)*exp(2)^2/(5*x^6-30*x^5+55*x^4+30*x^3-175*x^2+60*x+180)/exp(x),x,method=_RETURNVERBOS
E)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{180+60 x-175 x^2+30 x^3+55 x^4-30 x^5+5 x^6} \, dx=-\frac {x e^{\left (-x + 4\right )}}{5 \, {\left (x^{3} - 3 \, x^{2} + x + 6\right )}} \]

[In]

integrate((x^4-x^3-2*x^2+6*x-6)*exp(2)^2/(5*x^6-30*x^5+55*x^4+30*x^3-175*x^2+60*x+180)/exp(x),x, algorithm="fr
icas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{180+60 x-175 x^2+30 x^3+55 x^4-30 x^5+5 x^6} \, dx=- \frac {x e^{4} e^{- x}}{5 x^{3} - 15 x^{2} + 5 x + 30} \]

[In]

integrate((x**4-x**3-2*x**2+6*x-6)*exp(2)**2/(5*x**6-30*x**5+55*x**4+30*x**3-175*x**2+60*x+180)/exp(x),x)

[Out]

-x*exp(4)*exp(-x)/(5*x**3 - 15*x**2 + 5*x + 30)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{180+60 x-175 x^2+30 x^3+55 x^4-30 x^5+5 x^6} \, dx=-\frac {x e^{\left (-x + 4\right )}}{5 \, {\left (x^{3} - 3 \, x^{2} + x + 6\right )}} \]

[In]

integrate((x^4-x^3-2*x^2+6*x-6)*exp(2)^2/(5*x^6-30*x^5+55*x^4+30*x^3-175*x^2+60*x+180)/exp(x),x, algorithm="ma
xima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{180+60 x-175 x^2+30 x^3+55 x^4-30 x^5+5 x^6} \, dx=-\frac {{\left (x - 4\right )} e^{\left (-x + 4\right )} + 4 \, e^{\left (-x + 4\right )}}{5 \, {\left ({\left (x - 4\right )}^{3} + 9 \, {\left (x - 4\right )}^{2} + 25 \, x - 74\right )}} \]

[In]

integrate((x^4-x^3-2*x^2+6*x-6)*exp(2)^2/(5*x^6-30*x^5+55*x^4+30*x^3-175*x^2+60*x+180)/exp(x),x, algorithm="gi
ac")

[Out]

-1/5*((x - 4)*e^(-x + 4) + 4*e^(-x + 4))/((x - 4)^3 + 9*(x - 4)^2 + 25*x - 74)

Mupad [B] (verification not implemented)

Time = 8.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{4-x} \left (-6+6 x-2 x^2-x^3+x^4\right )}{180+60 x-175 x^2+30 x^3+55 x^4-30 x^5+5 x^6} \, dx=-\frac {x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^4}{5\,\left (x^3-3\,x^2+x+6\right )} \]

[In]

int(-(exp(-x)*exp(4)*(2*x^2 - 6*x + x^3 - x^4 + 6))/(60*x - 175*x^2 + 30*x^3 + 55*x^4 - 30*x^5 + 5*x^6 + 180),
x)

[Out]

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