\(\int \frac {e^{\frac {3-27 x+e^{19/5} x}{-22+e^{19/5}}} (-27+e^{19/5})}{-22+e^{19/5}} \, dx\) [3210]

Optimal result

Integrand size = 41, antiderivative size = 19 \[ \int \frac {e^{\frac {3-27 x+e^{19/5} x}{-22+e^{19/5}}} \left (-27+e^{19/5}\right )}{-22+e^{19/5}} \, dx=e^{\frac {3-5 x}{-22+e^{19/5}}+x} \]

[Out]

exp(x+(3-5*x)/(exp(19/5)-22))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {12, 2259, 2225} \[ \int \frac {e^{\frac {3-27 x+e^{19/5} x}{-22+e^{19/5}}} \left (-27+e^{19/5}\right )}{-22+e^{19/5}} \, dx=e^{-\frac {3-\left (27-e^{19/5}\right ) x}{22-e^{19/5}}} \]

[In]

Int[(E^((3 - 27*x + E^(19/5)*x)/(-22 + E^(19/5)))*(-27 + E^(19/5)))/(-22 + E^(19/5)),x]

[Out]

E^(-((3 - (27 - E^(19/5))*x)/(22 - E^(19/5))))

Rule 12

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

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2259

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (27-e^{19/5}\right ) \int e^{\frac {3-27 x+e^{19/5} x}{-22+e^{19/5}}} \, dx}{22-e^{19/5}} \\ & = \frac {\left (27-e^{19/5}\right ) \int e^{\frac {3-\left (27-e^{19/5}\right ) x}{-22+e^{19/5}}} \, dx}{22-e^{19/5}} \\ & = e^{-\frac {3-\left (27-e^{19/5}\right ) x}{22-e^{19/5}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\frac {3-27 x+e^{19/5} x}{-22+e^{19/5}}} \left (-27+e^{19/5}\right )}{-22+e^{19/5}} \, dx=e^{\frac {3+\left (-27+e^{19/5}\right ) x}{-22+e^{19/5}}} \]

[In]

Integrate[(E^((3 - 27*x + E^(19/5)*x)/(-22 + E^(19/5)))*(-27 + E^(19/5)))/(-22 + E^(19/5)),x]

[Out]

E^((3 + (-27 + E^(19/5))*x)/(-22 + E^(19/5)))

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
gosper	\({\mathrm e}^{\frac {x \,{\mathrm e}^{\frac {19}{5}}-27 x +3}{{\mathrm e}^{\frac {19}{5}}-22}}\)	\(18\)
derivativedivides	\({\mathrm e}^{\frac {x \,{\mathrm e}^{\frac {19}{5}}-27 x +3}{{\mathrm e}^{\frac {19}{5}}-22}}\)	\(18\)
default	\({\mathrm e}^{\frac {x \,{\mathrm e}^{\frac {19}{5}}-27 x +3}{{\mathrm e}^{\frac {19}{5}}-22}}\)	\(18\)
norman	\({\mathrm e}^{\frac {x \,{\mathrm e}^{\frac {19}{5}}-27 x +3}{{\mathrm e}^{\frac {19}{5}}-22}}\)	\(18\)
risch	\({\mathrm e}^{\frac {x \,{\mathrm e}^{\frac {19}{5}}-27 x +3}{{\mathrm e}^{\frac {19}{5}}-22}}\)	\(18\)
parts	\({\mathrm e}^{\frac {x \,{\mathrm e}^{\frac {19}{5}}-27 x +3}{{\mathrm e}^{\frac {19}{5}}-22}}\)	\(18\)
parallelrisch	\(\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{\frac {19}{5}}-27 x +3}{{\mathrm e}^{\frac {19}{5}}-22}} {\mathrm e}^{\frac {19}{5}}-22 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{\frac {19}{5}}-27 x +3}{{\mathrm e}^{\frac {19}{5}}-22}}}{{\mathrm e}^{\frac {19}{5}}-22}\)	\(48\)
meijerg	\(-\frac {{\mathrm e}^{\frac {19}{5}+\frac {3}{{\mathrm e}^{\frac {19}{5}}-22}} \left (1-{\mathrm e}^{\frac {x \left ({\mathrm e}^{\frac {19}{5}}-27\right )}{{\mathrm e}^{\frac {19}{5}}-22}}\right )}{{\mathrm e}^{\frac {19}{5}}-27}+\frac {27 \,{\mathrm e}^{\frac {3}{{\mathrm e}^{\frac {19}{5}}-22}} \left (1-{\mathrm e}^{\frac {x \left ({\mathrm e}^{\frac {19}{5}}-27\right )}{{\mathrm e}^{\frac {19}{5}}-22}}\right )}{{\mathrm e}^{\frac {19}{5}}-27}\)	\(72\)

[In]

int((exp(19/5)-27)*exp((x*exp(19/5)-27*x+3)/(exp(19/5)-22))/(exp(19/5)-22),x,method=_RETURNVERBOSE)

[Out]

exp((x*exp(19/5)-27*x+3)/(exp(19/5)-22))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {3-27 x+e^{19/5} x}{-22+e^{19/5}}} \left (-27+e^{19/5}\right )}{-22+e^{19/5}} \, dx=e^{\left (\frac {x e^{\frac {19}{5}} - 27 \, x + 3}{e^{\frac {19}{5}} - 22}\right )} \]

[In]

integrate((exp(19/5)-27)*exp((x*exp(19/5)-27*x+3)/(exp(19/5)-22))/(exp(19/5)-22),x, algorithm="fricas")

[Out]

e^((x*e^(19/5) - 27*x + 3)/(e^(19/5) - 22))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {3-27 x+e^{19/5} x}{-22+e^{19/5}}} \left (-27+e^{19/5}\right )}{-22+e^{19/5}} \, dx=e^{\frac {- 27 x + x e^{\frac {19}{5}} + 3}{-22 + e^{\frac {19}{5}}}} \]

[In]

integrate((exp(19/5)-27)*exp((x*exp(19/5)-27*x+3)/(exp(19/5)-22))/(exp(19/5)-22),x)

[Out]

exp((-27*x + x*exp(19/5) + 3)/(-22 + exp(19/5)))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {3-27 x+e^{19/5} x}{-22+e^{19/5}}} \left (-27+e^{19/5}\right )}{-22+e^{19/5}} \, dx=e^{\left (\frac {x e^{\frac {19}{5}} - 27 \, x + 3}{e^{\frac {19}{5}} - 22}\right )} \]

[In]

integrate((exp(19/5)-27)*exp((x*exp(19/5)-27*x+3)/(exp(19/5)-22))/(exp(19/5)-22),x, algorithm="maxima")

[Out]

e^((x*e^(19/5) - 27*x + 3)/(e^(19/5) - 22))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {3-27 x+e^{19/5} x}{-22+e^{19/5}}} \left (-27+e^{19/5}\right )}{-22+e^{19/5}} \, dx=e^{\left (\frac {x e^{\frac {19}{5}} - 27 \, x + 3}{e^{\frac {19}{5}} - 22}\right )} \]

[In]

integrate((exp(19/5)-27)*exp((x*exp(19/5)-27*x+3)/(exp(19/5)-22))/(exp(19/5)-22),x, algorithm="giac")

[Out]

e^((x*e^(19/5) - 27*x + 3)/(e^(19/5) - 22))

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \frac {e^{\frac {3-27 x+e^{19/5} x}{-22+e^{19/5}}} \left (-27+e^{19/5}\right )}{-22+e^{19/5}} \, dx={\mathrm {e}}^{-\frac {27\,x}{{\mathrm {e}}^{19/5}-22}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{19/5}}{{\mathrm {e}}^{19/5}-22}}\,{\mathrm {e}}^{\frac {3}{{\mathrm {e}}^{19/5}-22}} \]

[In]

int((exp((x*exp(19/5) - 27*x + 3)/(exp(19/5) - 22))*(exp(19/5) - 27))/(exp(19/5) - 22),x)

[Out]

exp(-(27*x)/(exp(19/5) - 22))*exp((x*exp(19/5))/(exp(19/5) - 22))*exp(3/(exp(19/5) - 22))