Integrand size = 38, antiderivative size = 18 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {e^{-2+\frac {e^5}{x}} (3+x)}{x^3} \]
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {e^{-2+\frac {e^5}{x}} (3+x)}{x^3} \]
Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(18)=36\).
Time = 0.20 (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 = {2726}
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.
\(\displaystyle \int \frac {e^{-\frac {2 x-e^5}{x}} \left (-2 x^2-9 x+e^5 (-x-3)\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {e^{\frac {e^5-2 x}{x}+5} (x+3)}{\left (\frac {e^5-2 x}{x^2}+\frac {2}{x}\right ) x^5}\) |
3.17.49.3.1 Defintions of rubi rules used
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]
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 )\) | \(653\) |
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {{\left (x + 3\right )} e^{\left (-\frac {2 \, x - e^{5}}{x}\right )}}{x^{3}} \]
Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {\left (x + 3\right ) e^{- \frac {2 x - e^{5}}{x}}}{x^{3}} \]
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 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=-3 \, e^{\left (-17\right )} \Gamma \left (4, -\frac {e^{5}}{x}\right ) + e^{\left (-12\right )} \Gamma \left (3, -\frac {e^{5}}{x}\right ) + 9 \, e^{\left (-17\right )} \Gamma \left (3, -\frac {e^{5}}{x}\right ) - 2 \, e^{\left (-12\right )} \Gamma \left (2, -\frac {e^{5}}{x}\right ) \]
-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)
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 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx={\left (\frac {{\left (2 \, x - e^{5}\right )}^{2} e^{\left (-\frac {2 \, x - e^{5}}{x} + 10\right )}}{x^{2}} - \frac {4 \, {\left (2 \, x - e^{5}\right )} e^{\left (-\frac {2 \, x - e^{5}}{x} + 10\right )}}{x} - \frac {3 \, {\left (2 \, x - e^{5}\right )}^{3} e^{\left (-\frac {2 \, x - e^{5}}{x} + 5\right )}}{x^{3}} + \frac {18 \, {\left (2 \, x - e^{5}\right )}^{2} e^{\left (-\frac {2 \, x - e^{5}}{x} + 5\right )}}{x^{2}} - \frac {36 \, {\left (2 \, x - e^{5}\right )} e^{\left (-\frac {2 \, x - e^{5}}{x} + 5\right )}}{x} + 4 \, e^{\left (-\frac {2 \, x - e^{5}}{x} + 10\right )} + 24 \, e^{\left (-\frac {2 \, x - e^{5}}{x} + 5\right )}\right )} e^{\left (-20\right )} \]
((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)
Time = 8.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-\frac {-e^5+2 x}{x}} \left (e^5 (-3-x)-9 x-2 x^2\right )}{x^5} \, dx=\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^5}{x}-2}\,\left (x+3\right )}{x^3} \]