3.67.78 $\int \frac{e^{\frac{20+x}{x}}(-40-2x)}{x^3}dx$

Optimal. Leaf size=24 $$\frac{2(e^{\frac{20+x}{x}}+(3-e^{32})x)}{x}$$

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Rubi [A]  time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.17, number of steps used = 1, number of rules used = 1, integrand size = 18, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.056, Rules used = {2288} $$\begin{array}{c}-\frac{40e^{\frac{x+20}{x}}}{x^3(\frac{1}{x}-\frac{x+20}{x^2})}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^((20 + x)/x)*(-40 - 2*x))/x^3,x]

[Out]

(-40*E^((20 + x)/x))/(x^3*(x^(-1) - (20 + x)/x^2))

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =-\frac{40e^{\frac{20+x}{x}}}{x^3(\frac{1}{x}-\frac{20+x}{x^2})}\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.58 $$\begin{array}{c}\frac{2e^{1+\frac{20}{x}}}{x}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^((20 + x)/x)*(-40 - 2*x))/x^3,x]

[Out]

(2*E^(1 + 20/x))/x

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fricas [A]  time = 0.46, size = 13, normalized size = 0.54 $$\begin{array}{c}\frac{2e^{\left(\frac{x+20}{x}\right)}}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-40)*exp((20+x)/x)/x^3,x, algorithm="fricas")

[Out]

2*e^((x + 20)/x)/x

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giac [A]  time = 0.19, size = 27, normalized size = 1.12 $$\begin{array}{c}\frac{(x+20)e^{\left(\frac{x+20}{x}\right)}}{10x}-\frac{1}{10}{e}^{\left(\frac{x+20}{x}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-40)*exp((20+x)/x)/x^3,x, algorithm="giac")

[Out]

1/10*(x + 20)*e^((x + 20)/x)/x - 1/10*e^((x + 20)/x)

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maple [A]  time = 0.06, size = 14, normalized size = 0.58

method	result	size
gosper	$\frac{2{\mathrm{e}}^{\frac{20+x}{x}}}{x}$	$14$
norman	$\frac{2{\mathrm{e}}^{\frac{20+x}{x}}}{x}$	$14$
risch	$\frac{2{\mathrm{e}}^{\frac{20+x}{x}}}{x}$	$14$
derivativedivides	$\frac{(1+\frac{20}{x}){\mathrm{e}}^{1+\frac{20}{x}}}{10}-\frac{{\mathrm{e}}^{1+\frac{20}{x}}}{10}$	$29$
default	$\frac{(1+\frac{20}{x}){\mathrm{e}}^{1+\frac{20}{x}}}{10}-\frac{{\mathrm{e}}^{1+\frac{20}{x}}}{10}$	$29$
meijerg	$-\frac{\mathrm{e}(1-{\mathrm{e}}^{\frac{20}{x}})}{10}+\frac{\mathrm{e}(1-\frac{(2-\frac{40}{x}){\mathrm{e}}^{\frac{20}{x}}}{2})}{10}$	$37$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x-40)*exp((20+x)/x)/x^3,x,method=_RETURNVERBOSE)

[Out]

2*exp((20+x)/x)/x

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maxima [C]  time = 0.46, size = 22, normalized size = 0.92 $$\begin{array}{c}-\frac{1}{10}e\mathrm{\Gamma }(2,-\frac{20}{x})+\frac{1}{10}{e}^{(\frac{20}{x}+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-40)*exp((20+x)/x)/x^3,x, algorithm="maxima")

[Out]

-1/10*e*gamma(2, -20/x) + 1/10*e^(20/x + 1)

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mupad [B]  time = 4.12, size = 13, normalized size = 0.54 $$\begin{array}{c}\frac{2\mathrm{e}{\mathrm{e}}^{20/x}}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x + 20)/x)*(2*x + 40))/x^3,x)

[Out]

(2*exp(1)*exp(20/x))/x

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sympy [A]  time = 0.11, size = 8, normalized size = 0.33 $$\begin{array}{c}\frac{2e^{\frac{x+20}{x}}}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-40)*exp((20+x)/x)/x**3,x)

[Out]

2*exp((x + 20)/x)/x

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