3.40.24 \(\int \frac {e^{.-\frac {8}{5}/x} (8-5 x)+45 x}{15 x^3} \, dx\)

Optimal. Leaf size=21 \[ 5+\frac {-3+\frac {1}{3} e^{\left .-\frac {8}{5}\right /x}}{x} \]

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Rubi [A]  time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 14, 2288} \begin {gather*} \frac {e^{\left .-\frac {8}{5}\right /x}}{3 x}-\frac {3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((8 - 5*x)/E^(8/(5*x)) + 45*x)/(15*x^3),x]

[Out]

-3/x + 1/(3*E^(8/(5*x))*x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{15} \int \frac {e^{\left .-\frac {8}{5}\right /x} (8-5 x)+45 x}{x^3} \, dx\\ &=\frac {1}{15} \int \left (\frac {45}{x^2}-\frac {e^{\left .-\frac {8}{5}\right /x} (-8+5 x)}{x^3}\right ) \, dx\\ &=-\frac {3}{x}-\frac {1}{15} \int \frac {e^{\left .-\frac {8}{5}\right /x} (-8+5 x)}{x^3} \, dx\\ &=-\frac {3}{x}+\frac {e^{\left .-\frac {8}{5}\right /x}}{3 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 18, normalized size = 0.86 \begin {gather*} \frac {-9+e^{\left .-\frac {8}{5}\right /x}}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((8 - 5*x)/E^(8/(5*x)) + 45*x)/(15*x^3),x]

[Out]

(-9 + E^(-8/(5*x)))/(3*x)

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fricas [A]  time = 0.57, size = 13, normalized size = 0.62 \begin {gather*} \frac {e^{\left (-\frac {8}{5 \, x}\right )} - 9}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*((-5*x+8)*exp(-8/5/x)+45*x)/x^3,x, algorithm="fricas")

[Out]

1/3*(e^(-8/5/x) - 9)/x

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giac [A]  time = 0.12, size = 17, normalized size = 0.81 \begin {gather*} \frac {e^{\left (-\frac {8}{5 \, x}\right )}}{3 \, x} - \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*((-5*x+8)*exp(-8/5/x)+45*x)/x^3,x, algorithm="giac")

[Out]

1/3*e^(-8/5/x)/x - 3/x

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maple [A]  time = 0.05, size = 18, normalized size = 0.86

method	result	size
derivativedivides	\(-\frac {3}{x}+\frac {{\mathrm e}^{-\frac {8}{5 x}}}{3 x}\)	\(18\)
default	\(-\frac {3}{x}+\frac {{\mathrm e}^{-\frac {8}{5 x}}}{3 x}\)	\(18\)
norman	\(\frac {-3 x +\frac {{\mathrm e}^{-\frac {8}{5 x}} x}{3}}{x^{2}}\)	\(18\)
risch	\(-\frac {3}{x}+\frac {{\mathrm e}^{-\frac {8}{5 x}}}{3 x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/15*((-5*x+8)*exp(-8/5/x)+45*x)/x^3,x,method=_RETURNVERBOSE)

[Out]

-3/x+1/3*exp(-8/5/x)/x

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maxima [C]  time = 0.36, size = 23, normalized size = 1.10 \begin {gather*} -\frac {3}{x} - \frac {5}{24} \, e^{\left (-\frac {8}{5 \, x}\right )} + \frac {5}{24} \, \Gamma \left (2, \frac {8}{5 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*((-5*x+8)*exp(-8/5/x)+45*x)/x^3,x, algorithm="maxima")

[Out]

-3/x - 5/24*e^(-8/5/x) + 5/24*gamma(2, 8/5/x)

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mupad [B]  time = 2.33, size = 13, normalized size = 0.62 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {8}{5\,x}}-9}{3\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - (exp(-8/(5*x))*(5*x - 8))/15)/x^3,x)

[Out]

(exp(-8/(5*x)) - 9)/(3*x)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.67 \begin {gather*} - \frac {3}{x} + \frac {e^{- \frac {8}{5 x}}}{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*((-5*x+8)*exp(-8/5/x)+45*x)/x**3,x)

[Out]

-3/x + exp(-8/(5*x))/(3*x)

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