3.79.96 \(\int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{6 x^2} \, dx\)

Optimal. Leaf size=15 \[ \frac {e^{13+\frac {23 x}{6}}+x}{x} \]

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {12, 2197} \begin {gather*} \frac {e^{\frac {1}{6} (23 x+60)+3}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(3 + (60 + 23*x)/6)*(-6 + 23*x))/(6*x^2),x]

[Out]

E^(3 + (60 + 23*x)/6)/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 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}{6} \int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{x^2} \, dx\\ &=\frac {e^{3+\frac {1}{6} (60+23 x)}}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.87 \begin {gather*} \frac {e^{13+\frac {23 x}{6}}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3 + (60 + 23*x)/6)*(-6 + 23*x))/(6*x^2),x]

[Out]

E^(13 + (23*x)/6)/x

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fricas [A]  time = 0.90, size = 10, normalized size = 0.67 \begin {gather*} \frac {e^{\left (\frac {23}{6} \, x + 13\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(23*x-6)*exp(3)*exp(23/12*x+5)^2/x^2,x, algorithm="fricas")

[Out]

e^(23/6*x + 13)/x

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giac [A]  time = 0.23, size = 10, normalized size = 0.67 \begin {gather*} \frac {e^{\left (\frac {23}{6} \, x + 13\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(23*x-6)*exp(3)*exp(23/12*x+5)^2/x^2,x, algorithm="giac")

[Out]

e^(23/6*x + 13)/x

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maple [A]  time = 0.08, size = 11, normalized size = 0.73

method	result	size
risch	\(\frac {{\mathrm e}^{13+\frac {23 x}{6}}}{x}\)	\(11\)
gosper	\(\frac {{\mathrm e}^{3} {\mathrm e}^{\frac {23 x}{6}+10}}{x}\)	\(15\)
derivativedivides	\(\frac {{\mathrm e}^{3} {\mathrm e}^{\frac {23 x}{6}+10}}{x}\)	\(15\)
default	\(\frac {{\mathrm e}^{3} {\mathrm e}^{\frac {23 x}{6}+10}}{x}\)	\(15\)
norman	\(\frac {{\mathrm e}^{3} {\mathrm e}^{\frac {23 x}{6}+10}}{x}\)	\(15\)
meijerg	\(\frac {23 \,{\mathrm e}^{23-\frac {23 x \,{\mathrm e}^{10}}{6}+\frac {23 x}{6}} \left (\frac {6 \,{\mathrm e}^{-10}}{23 x}-9-\ln \relax (x )-\ln \left (23\right )+\ln \relax (2)+\ln \relax (3)-i \pi -\frac {3 \,{\mathrm e}^{-10} \left (2+\frac {23 x \,{\mathrm e}^{10}}{3}\right )}{23 x}+\frac {6 \,{\mathrm e}^{-10+\frac {23 x \,{\mathrm e}^{10}}{6}}}{23 x}+\ln \left (-\frac {23 x \,{\mathrm e}^{10}}{6}\right )+\expIntegralEi \left (1, -\frac {23 x \,{\mathrm e}^{10}}{6}\right )\right )}{6}+\frac {23 \,{\mathrm e}^{13-\frac {23 x \,{\mathrm e}^{10}}{6}+\frac {23 x}{6}} \left (\ln \relax (x )+\ln \left (23\right )-\ln \relax (2)-\ln \relax (3)+10+i \pi -\ln \left (-\frac {23 x \,{\mathrm e}^{10}}{6}\right )-\expIntegralEi \left (1, -\frac {23 x \,{\mathrm e}^{10}}{6}\right )\right )}{6}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/6*(23*x-6)*exp(3)*exp(23/12*x+5)^2/x^2,x,method=_RETURNVERBOSE)

[Out]

1/x*exp(13+23/6*x)

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maxima [C]  time = 0.40, size = 18, normalized size = 1.20 \begin {gather*} \frac {23}{6} \, {\rm Ei}\left (\frac {23}{6} \, x\right ) e^{13} - \frac {23}{6} \, e^{13} \Gamma \left (-1, -\frac {23}{6} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(23*x-6)*exp(3)*exp(23/12*x+5)^2/x^2,x, algorithm="maxima")

[Out]

23/6*Ei(23/6*x)*e^13 - 23/6*e^13*gamma(-1, -23/6*x)

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mupad [B]  time = 0.08, size = 10, normalized size = 0.67 \begin {gather*} \frac {{\mathrm {e}}^{\frac {23\,x}{6}}\,{\mathrm {e}}^{13}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3)*exp((23*x)/6 + 10)*(23*x - 6))/(6*x^2),x)

[Out]

(exp((23*x)/6)*exp(13))/x

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sympy [A]  time = 0.11, size = 12, normalized size = 0.80 \begin {gather*} \frac {e^{3} e^{\frac {23 x}{6} + 10}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(23*x-6)*exp(3)*exp(23/12*x+5)**2/x**2,x)

[Out]

exp(3)*exp(23*x/6 + 10)/x

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