\(\int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{6 x^2} \, dx\) [7896]

Optimal result

Integrand size = 25, antiderivative size = 15 \[ \int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{6 x^2} \, dx=\frac {e^{13+\frac {23 x}{6}}+x}{x} \]

[Out]

(x+exp(23/12*x+5)^2*exp(3))/x

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {12, 2228} \[ \int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{6 x^2} \, dx=\frac {e^{\frac {1}{6} (23 x+60)+3}}{x} \]

[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 2228

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{6 x^2} \, dx=\frac {e^{13+\frac {23 x}{6}}}{x} \]

[In]

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

[Out]

E^(13 + (23*x)/6)/x

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {23 x}{6}+13}}{x}\)	\(11\)
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	\(\frac {{\mathrm e}^{3} {\mathrm e}^{\frac {23 x}{6}+10}}{x}\)	\(15\)
parallelrisch	\(\frac {{\mathrm e}^{3} {\mathrm e}^{\frac {23 x}{6}+10}}{x}\)	\(15\)
meijerg	\(\frac {23 \,{\mathrm e}^{23+\frac {23 x}{6}-\frac {23 x \,{\mathrm e}^{10}}{6}} \left (\frac {6 \,{\mathrm e}^{-10}}{23 x}-9-\ln \left (x \right )-\ln \left (23\right )+\ln \left (2\right )+\ln \left (3\right )-i \pi -\frac {3 \,{\mathrm e}^{-10} \left (2+\frac {23 x \,{\mathrm e}^{10}}{3}\right )}{23 x}+\frac {6 \,{\mathrm e}^{-10+\frac {23 x \,{\mathrm e}^{10}}{6}}}{23 x}+\ln \left (-\frac {23 x \,{\mathrm e}^{10}}{6}\right )+\operatorname {Ei}_{1}\left (-\frac {23 x \,{\mathrm e}^{10}}{6}\right )\right )}{6}+\frac {23 \,{\mathrm e}^{\frac {23 x}{6}+13-\frac {23 x \,{\mathrm e}^{10}}{6}} \left (\ln \left (x \right )+\ln \left (23\right )-\ln \left (2\right )-\ln \left (3\right )+10+i \pi -\ln \left (-\frac {23 x \,{\mathrm e}^{10}}{6}\right )-\operatorname {Ei}_{1}\left (-\frac {23 x \,{\mathrm e}^{10}}{6}\right )\right )}{6}\)	\(128\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{6 x^2} \, dx=\frac {e^{\left (\frac {23}{6} \, x + 13\right )}}{x} \]

[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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{6 x^2} \, dx=\frac {e^{3} e^{\frac {23 x}{6} + 10}}{x} \]

[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

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{6 x^2} \, dx=\frac {23}{6} \, {\rm Ei}\left (\frac {23}{6} \, x\right ) e^{13} - \frac {23}{6} \, e^{13} \Gamma \left (-1, -\frac {23}{6} \, x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{6 x^2} \, dx=\frac {e^{\left (\frac {23}{6} \, x + 13\right )}}{x} \]

[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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {e^{3+\frac {1}{6} (60+23 x)} (-6+23 x)}{6 x^2} \, dx=\frac {{\mathrm {e}}^{\frac {23\,x}{6}}\,{\mathrm {e}}^{13}}{x} \]

[In]

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

[Out]

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