\(\int \frac {e^{-\frac {-e^5+2 x}{x}} (e^5 (-3-x)-9 x-2 x^2)}{x^5} \, dx\) [1649]

Optimal result

Integrand size = 38, antiderivative size = 18 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {e^{-2+\frac {e^5}{x}} (3+x)}{x^3} \]

[Out]

(3+x)/x^3/exp(2-exp(5)/x)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(18)=36\).

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.28, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2326} \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {e^{\frac {e^5-2 x}{x}+5} (x+3)}{\left (\frac {e^5-2 x}{x^2}+\frac {2}{x}\right ) x^5} \]

[In]

Int[(E^5*(-3 - x) - 9*x - 2*x^2)/(E^((-E^5 + 2*x)/x)*x^5),x]

[Out]

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

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{5+\frac {e^5-2 x}{x}} (3+x)}{\left (\frac {e^5-2 x}{x^2}+\frac {2}{x}\right ) x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {e^{-2+\frac {e^5}{x}} (3+x)}{x^3} \]

[In]

Integrate[(E^5*(-3 - x) - 9*x - 2*x^2)/(E^((-E^5 + 2*x)/x)*x^5),x]

[Out]

(E^(-2 + E^5/x)*(3 + x))/x^3

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06

method	result	size
risch	\(\frac {\left (3+x \right ) {\mathrm e}^{\frac {{\mathrm e}^{5}-2 x}{x}}}{x^{3}}\)	\(19\)
gosper	\(\frac {\left (3+x \right ) {\mathrm e}^{\frac {{\mathrm e}^{5}-2 x}{x}}}{x^{3}}\)	\(22\)
parallelrisch	\(\frac {\left (3+x \right ) {\mathrm e}^{\frac {{\mathrm e}^{5}-2 x}{x}}}{x^{3}}\)	\(22\)
norman	\(\frac {\left (x^{2}+3 x \right ) {\mathrm e}^{-\frac {-{\mathrm e}^{5}+2 x}{x}}}{x^{4}}\)	\(27\)
meijerg	\(2 \,{\mathrm e}^{-8+\frac {{\mathrm e}^{5}}{x}-\frac {{\mathrm e}^{3}}{x}} \left (1-\frac {\left (2-\frac {2 \,{\mathrm e}^{3}}{x}\right ) {\mathrm e}^{\frac {{\mathrm e}^{3}}{x}}}{2}\right )+{\mathrm e}^{-11+\frac {{\mathrm e}^{5}}{x}-\frac {{\mathrm e}^{3}}{x}} \left (-{\mathrm e}^{5}-9\right ) \left (2-\frac {\left (\frac {3 \,{\mathrm e}^{6}}{x^{2}}-\frac {6 \,{\mathrm e}^{3}}{x}+6\right ) {\mathrm e}^{\frac {{\mathrm e}^{3}}{x}}}{3}\right )+3 \,{\mathrm e}^{-9+\frac {{\mathrm e}^{5}}{x}-\frac {{\mathrm e}^{3}}{x}} \left (6-\frac {\left (-\frac {4 \,{\mathrm e}^{9}}{x^{3}}+\frac {12 \,{\mathrm e}^{6}}{x^{2}}-\frac {24 \,{\mathrm e}^{3}}{x}+24\right ) {\mathrm e}^{\frac {{\mathrm e}^{3}}{x}}}{4}\right )\)	\(142\)
derivativedivides	\(-{\mathrm e}^{-20} \left ({\mathrm e}^{10} \left (-\left (\left (2-\frac {{\mathrm e}^{5}}{x}\right )^{2}-\frac {4 \,{\mathrm e}^{5}}{x}+16\right ) {\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}-8 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )-24 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} {\mathrm e}^{5}+16 \,{\mathrm e}^{10} \left ({\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} x \,{\mathrm e}^{-5}+{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )+72 \,{\mathrm e}^{5} {\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )+8 \,{\mathrm e}^{10} {\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )-36 \,{\mathrm e}^{5} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}\right )+18 \,{\mathrm e}^{5} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )^{2}-2 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )-2 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}\right )-3 \,{\mathrm e}^{5} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )^{3}-3 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )^{2}-6 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )-6 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}\right )-24 \,{\mathrm e}^{10} \left (2 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} x \,{\mathrm e}^{-5}+{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )+12 \,{\mathrm e}^{10} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}+4 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} x \,{\mathrm e}^{-5}\right )-2 \,{\mathrm e}^{10} \left (-\left (-\frac {{\mathrm e}^{5}}{x}+7\right ) {\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}+8 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} x \,{\mathrm e}^{-5}-4 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )+108 \,{\mathrm e}^{5} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}-2 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )-54 \,{\mathrm e}^{5} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (-\frac {{\mathrm e}^{5}}{x}+5\right )-4 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )+9 \,{\mathrm e}^{5} \left (-\left (\left (2-\frac {{\mathrm e}^{5}}{x}\right )^{2}-\frac {4 \,{\mathrm e}^{5}}{x}+16\right ) {\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}-8 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )+12 \,{\mathrm e}^{10} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}-2 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )-6 \,{\mathrm e}^{10} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (-\frac {{\mathrm e}^{5}}{x}+5\right )-4 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )\right )\)	\(653\)
default	\(-{\mathrm e}^{-20} \left ({\mathrm e}^{10} \left (-\left (\left (2-\frac {{\mathrm e}^{5}}{x}\right )^{2}-\frac {4 \,{\mathrm e}^{5}}{x}+16\right ) {\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}-8 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )-24 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} {\mathrm e}^{5}+16 \,{\mathrm e}^{10} \left ({\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} x \,{\mathrm e}^{-5}+{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )+72 \,{\mathrm e}^{5} {\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )+8 \,{\mathrm e}^{10} {\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )-36 \,{\mathrm e}^{5} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}\right )+18 \,{\mathrm e}^{5} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )^{2}-2 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )-2 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}\right )-3 \,{\mathrm e}^{5} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )^{3}-3 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )^{2}-6 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (2-\frac {{\mathrm e}^{5}}{x}\right )-6 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}\right )-24 \,{\mathrm e}^{10} \left (2 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} x \,{\mathrm e}^{-5}+{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )+12 \,{\mathrm e}^{10} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}+4 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} x \,{\mathrm e}^{-5}\right )-2 \,{\mathrm e}^{10} \left (-\left (-\frac {{\mathrm e}^{5}}{x}+7\right ) {\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}+8 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} x \,{\mathrm e}^{-5}-4 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )+108 \,{\mathrm e}^{5} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}-2 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )-54 \,{\mathrm e}^{5} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (-\frac {{\mathrm e}^{5}}{x}+5\right )-4 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )+9 \,{\mathrm e}^{5} \left (-\left (\left (2-\frac {{\mathrm e}^{5}}{x}\right )^{2}-\frac {4 \,{\mathrm e}^{5}}{x}+16\right ) {\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}-8 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )+12 \,{\mathrm e}^{10} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2}-2 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )-6 \,{\mathrm e}^{10} \left (-{\mathrm e}^{\frac {{\mathrm e}^{5}}{x}-2} \left (-\frac {{\mathrm e}^{5}}{x}+5\right )-4 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {{\mathrm e}^{5}}{x}\right )\right )\right )\)	\(653\)

[In]

int(((-3-x)*exp(5)-2*x^2-9*x)/x^5/exp((-exp(5)+2*x)/x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {{\left (x + 3\right )} e^{\left (-\frac {2 \, x - e^{5}}{x}\right )}}{x^{3}} \]

[In]

integrate(((-3-x)*exp(5)-2*x^2-9*x)/x^5/exp((-exp(5)+2*x)/x),x, algorithm="fricas")

[Out]

(x + 3)*e^(-(2*x - e^5)/x)/x^3

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {\left (x + 3\right ) e^{- \frac {2 x - e^{5}}{x}}}{x^{3}} \]

[In]

integrate(((-3-x)*exp(5)-2*x**2-9*x)/x**5/exp((-exp(5)+2*x)/x),x)

[Out]

(x + 3)*exp(-(2*x - exp(5))/x)/x**3

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.89 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=-3 \, e^{\left (-17\right )} \Gamma \left (4, -\frac {e^{5}}{x}\right ) + e^{\left (-12\right )} \Gamma \left (3, -\frac {e^{5}}{x}\right ) + 9 \, e^{\left (-17\right )} \Gamma \left (3, -\frac {e^{5}}{x}\right ) - 2 \, e^{\left (-12\right )} \Gamma \left (2, -\frac {e^{5}}{x}\right ) \]

[In]

integrate(((-3-x)*exp(5)-2*x^2-9*x)/x^5/exp((-exp(5)+2*x)/x),x, algorithm="maxima")

[Out]

-3*e^(-17)*gamma(4, -e^5/x) + e^(-12)*gamma(3, -e^5/x) + 9*e^(-17)*gamma(3, -e^5/x) - 2*e^(-12)*gamma(2, -e^5/
x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (16) = 32\).

Time = 0.26 (sec) , antiderivative size = 190, normalized size of antiderivative = 10.56 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx={\left (\frac {{\left (2 \, x - e^{5}\right )}^{2} e^{\left (-\frac {2 \, x - e^{5}}{x} + 10\right )}}{x^{2}} - \frac {4 \, {\left (2 \, x - e^{5}\right )} e^{\left (-\frac {2 \, x - e^{5}}{x} + 10\right )}}{x} - \frac {3 \, {\left (2 \, x - e^{5}\right )}^{3} e^{\left (-\frac {2 \, x - e^{5}}{x} + 5\right )}}{x^{3}} + \frac {18 \, {\left (2 \, x - e^{5}\right )}^{2} e^{\left (-\frac {2 \, x - e^{5}}{x} + 5\right )}}{x^{2}} - \frac {36 \, {\left (2 \, x - e^{5}\right )} e^{\left (-\frac {2 \, x - e^{5}}{x} + 5\right )}}{x} + 4 \, e^{\left (-\frac {2 \, x - e^{5}}{x} + 10\right )} + 24 \, e^{\left (-\frac {2 \, x - e^{5}}{x} + 5\right )}\right )} e^{\left (-20\right )} \]

[In]

integrate(((-3-x)*exp(5)-2*x^2-9*x)/x^5/exp((-exp(5)+2*x)/x),x, algorithm="giac")

[Out]

((2*x - e^5)^2*e^(-(2*x - e^5)/x + 10)/x^2 - 4*(2*x - e^5)*e^(-(2*x - e^5)/x + 10)/x - 3*(2*x - e^5)^3*e^(-(2*
x - e^5)/x + 5)/x^3 + 18*(2*x - e^5)^2*e^(-(2*x - e^5)/x + 5)/x^2 - 36*(2*x - e^5)*e^(-(2*x - e^5)/x + 5)/x +
4*e^(-(2*x - e^5)/x + 10) + 24*e^(-(2*x - e^5)/x + 5))*e^(-20)

Mupad [B] (verification not implemented)

Time = 8.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^5}{x}-2}\,\left (x+3\right )}{x^3} \]

[In]

int(-(exp(-(2*x - exp(5))/x)*(9*x + exp(5)*(x + 3) + 2*x^2))/x^5,x)

[Out]

(exp(exp(5)/x - 2)*(x + 3))/x^3