3.39.46 \(\int -\frac {2 e^{\frac {1}{3} (15-\log (x))}}{3 x} \, dx\)

Optimal. Leaf size=12 \[ 3+\frac {2 e^5}{\sqrt [3]{x}} \]

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Rubi [A]  time = 0.02, antiderivative size = 10, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {12, 2274, 30} \begin {gather*} \frac {2 e^5}{\sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*E^((15 - Log[x])/3))/(3*x),x]

[Out]

(2*E^5)/x^(1/3)

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (\frac {2}{3} \int \frac {e^{\frac {1}{3} (15-\log (x))}}{x} \, dx\right )\\ &=-\left (\frac {2}{3} \int \frac {e^5}{x^{4/3}} \, dx\right )\\ &=-\left (\frac {1}{3} \left (2 e^5\right ) \int \frac {1}{x^{4/3}} \, dx\right )\\ &=\frac {2 e^5}{\sqrt [3]{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.83 \begin {gather*} \frac {2 e^5}{\sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^((15 - Log[x])/3))/(3*x),x]

[Out]

(2*E^5)/x^(1/3)

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fricas [A]  time = 0.60, size = 7, normalized size = 0.58 \begin {gather*} \frac {2 \, e^{5}}{x^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*exp(-1/3*log(x)+5)/x,x, algorithm="fricas")

[Out]

2*e^5/x^(1/3)

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giac [A]  time = 0.19, size = 9, normalized size = 0.75 \begin {gather*} 2 \, e^{\left (-\frac {1}{3} \, \log \relax (x) + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*exp(-1/3*log(x)+5)/x,x, algorithm="giac")

[Out]

2*e^(-1/3*log(x) + 5)

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maple [A]  time = 0.02, size = 8, normalized size = 0.67

method	result	size
risch	\(\frac {2 \,{\mathrm e}^{5}}{x^{\frac {1}{3}}}\)	\(8\)
gosper	\(2 \,{\mathrm e}^{-\frac {\ln \relax (x )}{3}+5}\)	\(10\)
derivativedivides	\(2 \,{\mathrm e}^{-\frac {\ln \relax (x )}{3}+5}\)	\(10\)
default	\(2 \,{\mathrm e}^{-\frac {\ln \relax (x )}{3}+5}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2/3*exp(-1/3*ln(x)+5)/x,x,method=_RETURNVERBOSE)

[Out]

2*exp(5)/x^(1/3)

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maxima [A]  time = 0.36, size = 9, normalized size = 0.75 \begin {gather*} 2 \, e^{\left (-\frac {1}{3} \, \log \relax (x) + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*exp(-1/3*log(x)+5)/x,x, algorithm="maxima")

[Out]

2*e^(-1/3*log(x) + 5)

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mupad [B]  time = 2.26, size = 7, normalized size = 0.58 \begin {gather*} \frac {2\,{\mathrm {e}}^5}{x^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*exp(5 - log(x)/3))/(3*x),x)

[Out]

(2*exp(5))/x^(1/3)

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sympy [A]  time = 0.05, size = 8, normalized size = 0.67 \begin {gather*} \frac {2 e^{5}}{\sqrt [3]{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*exp(-1/3*ln(x)+5)/x,x)

[Out]

2*exp(5)/x**(1/3)

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