3.59.13 \(\int \frac {-120+e^{3-x} (20+20 x)}{x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac {20 \left (4+x+\frac {x+\left (1-e^{3-x}\right ) x}{x}\right )}{x} \]

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Rubi [A]  time = 0.04, antiderivative size = 18, normalized size of antiderivative = 0.67, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {14, 2197} \begin {gather*} \frac {120}{x}-\frac {20 e^{3-x}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-120 + E^(3 - x)*(20 + 20*x))/x^2,x]

[Out]

120/x - (20*E^(3 - x))/x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {120}{x^2}+\frac {20 e^{3-x} (1+x)}{x^2}\right ) \, dx\\ &=\frac {120}{x}+20 \int \frac {e^{3-x} (1+x)}{x^2} \, dx\\ &=\frac {120}{x}-\frac {20 e^{3-x}}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 16, normalized size = 0.59 \begin {gather*} \frac {20 \left (6-e^{3-x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-120 + E^(3 - x)*(20 + 20*x))/x^2,x]

[Out]

(20*(6 - E^(3 - x)))/x

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fricas [A]  time = 0.57, size = 13, normalized size = 0.48 \begin {gather*} -\frac {20 \, {\left (e^{\left (-x + 3\right )} - 6\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x+20)*exp(3-x)-120)/x^2,x, algorithm="fricas")

[Out]

-20*(e^(-x + 3) - 6)/x

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giac [A]  time = 0.22, size = 13, normalized size = 0.48 \begin {gather*} -\frac {20 \, {\left (e^{\left (-x + 3\right )} - 6\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x+20)*exp(3-x)-120)/x^2,x, algorithm="giac")

[Out]

-20*(e^(-x + 3) - 6)/x

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maple [A]  time = 0.07, size = 15, normalized size = 0.56

method	result	size
norman	\(\frac {120-20 \,{\mathrm e}^{3-x}}{x}\)	\(15\)
derivativedivides	\(\frac {120}{x}-\frac {20 \,{\mathrm e}^{3-x}}{x}\)	\(18\)
default	\(\frac {120}{x}-\frac {20 \,{\mathrm e}^{3-x}}{x}\)	\(18\)
risch	\(\frac {120}{x}-\frac {20 \,{\mathrm e}^{3-x}}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x+20)*exp(3-x)-120)/x^2,x,method=_RETURNVERBOSE)

[Out]

(120-20*exp(3-x))/x

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maxima [C]  time = 0.38, size = 21, normalized size = 0.78 \begin {gather*} 20 \, {\rm Ei}\left (-x\right ) e^{3} - 20 \, e^{3} \Gamma \left (-1, x\right ) + \frac {120}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x+20)*exp(3-x)-120)/x^2,x, algorithm="maxima")

[Out]

20*Ei(-x)*e^3 - 20*e^3*gamma(-1, x) + 120/x

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mupad [B]  time = 4.00, size = 15, normalized size = 0.56 \begin {gather*} -\frac {20\,{\mathrm {e}}^{3-x}-120}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3 - x)*(20*x + 20) - 120)/x^2,x)

[Out]

-(20*exp(3 - x) - 120)/x

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sympy [A]  time = 0.09, size = 10, normalized size = 0.37 \begin {gather*} - \frac {20 e^{3 - x}}{x} + \frac {120}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x+20)*exp(3-x)-120)/x**2,x)

[Out]

-20*exp(3 - x)/x + 120/x

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