3.81.15 \(\int \frac {e^{\frac {1}{8} (16-9 x)} (-24-9 x)}{4 x^4} \, dx\)

Optimal. Leaf size=14 \[ \frac {2 e^{2-\frac {9 x}{8}}}{x^3} \]

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Rubi [A]  time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 2197} \begin {gather*} \frac {2 e^{\frac {1}{8} (16-9 x)}}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((16 - 9*x)/8)*(-24 - 9*x))/(4*x^4),x]

[Out]

(2*E^((16 - 9*x)/8))/x^3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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} &=\frac {1}{4} \int \frac {e^{\frac {1}{8} (16-9 x)} (-24-9 x)}{x^4} \, dx\\ &=\frac {2 e^{\frac {1}{8} (16-9 x)}}{x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {2 e^{2-\frac {9 x}{8}}}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((16 - 9*x)/8)*(-24 - 9*x))/(4*x^4),x]

[Out]

(2*E^(2 - (9*x)/8))/x^3

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fricas [A]  time = 0.99, size = 11, normalized size = 0.79 \begin {gather*} \frac {2 \, e^{\left (-\frac {9}{8} \, x + 2\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-9*x-24)*exp(-9/8*x+2)/x^4,x, algorithm="fricas")

[Out]

2*e^(-9/8*x + 2)/x^3

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giac [A]  time = 0.17, size = 11, normalized size = 0.79 \begin {gather*} \frac {2 \, e^{\left (-\frac {9}{8} \, x + 2\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-9*x-24)*exp(-9/8*x+2)/x^4,x, algorithm="giac")

[Out]

2*e^(-9/8*x + 2)/x^3

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maple [A]  time = 0.12, size = 12, normalized size = 0.86

method	result	size
gosper	\(\frac {2 \,{\mathrm e}^{-\frac {9 x}{8}+2}}{x^{3}}\)	\(12\)
derivativedivides	\(\frac {2 \,{\mathrm e}^{-\frac {9 x}{8}+2}}{x^{3}}\)	\(12\)
default	\(\frac {2 \,{\mathrm e}^{-\frac {9 x}{8}+2}}{x^{3}}\)	\(12\)
norman	\(\frac {2 \,{\mathrm e}^{-\frac {9 x}{8}+2}}{x^{3}}\)	\(12\)
risch	\(\frac {2 \,{\mathrm e}^{-\frac {9 x}{8}+2}}{x^{3}}\)	\(12\)
meijerg	\(-\frac {2187 \,{\mathrm e}^{2} \left (-\frac {512}{2187 x^{3}}+\frac {32}{81 x^{2}}-\frac {4}{9 x}+\frac {11}{36}-\frac {\ln \relax (x )}{6}-\frac {\ln \relax (3)}{3}+\frac {\ln \relax (2)}{2}+\frac {-\frac {11}{36} x^{3}+\frac {4}{9} x^{2}-\frac {32}{81} x +\frac {512}{2187}}{x^{3}}-\frac {64 \left (\frac {81}{16} x^{2}-\frac {9}{2} x +8\right ) {\mathrm e}^{-\frac {9 x}{8}}}{2187 x^{3}}+\frac {\ln \left (\frac {9 x}{8}\right )}{6}+\frac {\expIntegralEi \left (1, \frac {9 x}{8}\right )}{6}\right )}{256}-\frac {729 \,{\mathrm e}^{2} \left (-\frac {32}{81 x^{2}}+\frac {8}{9 x}-\frac {3}{4}+\frac {\ln \relax (x )}{2}+\ln \relax (3)-\frac {3 \ln \relax (2)}{2}+\frac {\frac {3}{4} x^{2}-\frac {8}{9} x +\frac {32}{81}}{x^{2}}-\frac {32 \left (3-\frac {27 x}{8}\right ) {\mathrm e}^{-\frac {9 x}{8}}}{243 x^{2}}-\frac {\ln \left (\frac {9 x}{8}\right )}{2}-\frac {\expIntegralEi \left (1, \frac {9 x}{8}\right )}{2}\right )}{256}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-9*x-24)*exp(-9/8*x+2)/x^4,x,method=_RETURNVERBOSE)

[Out]

2/x^3*exp(-9/8*x+2)

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maxima [C]  time = 0.38, size = 19, normalized size = 1.36 \begin {gather*} \frac {729}{256} \, e^{2} \Gamma \left (-2, \frac {9}{8} \, x\right ) + \frac {2187}{256} \, e^{2} \Gamma \left (-3, \frac {9}{8} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-9*x-24)*exp(-9/8*x+2)/x^4,x, algorithm="maxima")

[Out]

729/256*e^2*gamma(-2, 9/8*x) + 2187/256*e^2*gamma(-3, 9/8*x)

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mupad [B]  time = 0.08, size = 11, normalized size = 0.79 \begin {gather*} \frac {2\,{\mathrm {e}}^{-\frac {9\,x}{8}}\,{\mathrm {e}}^2}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2 - (9*x)/8)*(9*x + 24))/(4*x^4),x)

[Out]

(2*exp(-(9*x)/8)*exp(2))/x^3

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sympy [A]  time = 0.09, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 e^{2 - \frac {9 x}{8}}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-9*x-24)*exp(-9/8*x+2)/x**4,x)

[Out]

2*exp(2 - 9*x/8)/x**3

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