3.17.49 $\int \frac{e^{-\frac{-e^5+2x}{x}}(e^5(-3-x)-9x-2x^2)}{x^5}dx$

Optimal. Leaf size=18 $$\frac{e^{-2+\frac{e^5}{x}}(3+x)}{x^3}$$

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Rubi [B]  time = 0.06, antiderivative size = 41, normalized size of antiderivative = 2.28, number of steps used = 1, number of rules used = 1, integrand size = 38, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.026, Rules used = {2288} $$\begin{array}{c}\frac{e^{\frac{e^5-2x}{x}+5}(x+3)}{(\frac{e^5-2x}{x^2}+\frac{2}{x})x^5}\end{array}$$

Antiderivative was successfully verified.

[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 2288

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{e^{5+\frac{e^5-2x}{x}}(3+x)}{(\frac{e^5-2x}{x^2}+\frac{2}{x})x^5}\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 $$\begin{array}{c}\frac{e^{-2+\frac{e^5}{x}}(3+x)}{x^3}\end{array}$$

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.74, size = 21, normalized size = 1.17 $$\begin{array}{c}\frac{(x+3)e^{(-\frac{2x-e^5}{x})}}{x^3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 0.28, size = 190, normalized size = 10.56 $$\begin{array}{c}(\frac{{(2x-e^5)}^2{e}^{(-\frac{2x-e^5}{x}+10)}}{x^2}-\frac{4(2x-e^5)e^{(-\frac{2x-e^5}{x}+10)}}{x}-\frac{3{(2x-e^5)}^3{e}^{(-\frac{2x-e^5}{x}+5)}}{x^3}+\frac{18{(2x-e^5)}^2{e}^{(-\frac{2x-e^5}{x}+5)}}{x^2}-\frac{36(2x-e^5)e^{(-\frac{2x-e^5}{x}+5)}}{x}+4e^{(-\frac{2x-e^5}{x}+10)}+24e^{(-\frac{2x-e^5}{x}+5)})e^{(-20)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.18, size = 19, normalized size = 1.06

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [C]  time = 0.46, size = 52, normalized size = 2.89 $$\begin{array}{c}-3e^{(-17)}\mathrm{\Gamma }(4,-\frac{e^5}{x})+e^{(-12)}\mathrm{\Gamma }(3,-\frac{e^5}{x})+9e^{(-17)}\mathrm{\Gamma }(3,-\frac{e^5}{x})-2e^{(-12)}\mathrm{\Gamma }(2,-\frac{e^5}{x})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 1.05, size = 16, normalized size = 0.89 $$\begin{array}{c}\frac{{\mathrm{e}}^{\frac{{\mathrm{e}}^5}{x}-2}(x+3)}{x^3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.13, size = 15, normalized size = 0.83 $$\begin{array}{c}\frac{(x+3)e^{-\frac{2x-e^5}{x}}}{x^3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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