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

Optimal. Leaf size=19 \[ e^{\frac {3-5 x}{-22+e^{19/5}}+x} \]

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Rubi [A]  time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.53, number of steps used = 3, number of rules used = 3, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {12, 2227, 2194} \begin {gather*} e^{-\frac {3-\left (27-e^{19/5}\right ) x}{22-e^{19/5}}} \end {gather*}

Antiderivative was successfully verified.

[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 2194

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 2227

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 23, normalized size = 1.21 \begin {gather*} e^{\frac {3+\left (-27+e^{19/5}\right ) x}{-22+e^{19/5}}} \end {gather*}

Antiderivative was successfully verified.

[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)))

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fricas [A]  time = 0.56, size = 17, normalized size = 0.89 \begin {gather*} e^{\left (\frac {x e^{\frac {19}{5}} - 27 \, x + 3}{e^{\frac {19}{5}} - 22}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.25, size = 17, normalized size = 0.89 \begin {gather*} e^{\left (\frac {x e^{\frac {19}{5}} - 27 \, x + 3}{e^{\frac {19}{5}} - 22}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.04, size = 18, normalized size = 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	\({\mathrm e}^{\frac {x \,{\mathrm e}^{\frac {19}{5}}-27 x +3}{{\mathrm e}^{\frac {19}{5}}-22}}\)	\(18\)
meijerg	\(-\frac {{\mathrm e}^{\frac {3}{{\mathrm e}^{\frac {19}{5}}-22}+\frac {19}{5}} \left (1-{\mathrm e}^{\frac {x \left ({\mathrm e}^{\frac {19}{5}}-27\right )}{{\mathrm e}^{\frac {19}{5}}-22}}\right )}{{\mathrm e}^{\frac {19}{5}}-27}+\frac {27 \,{\mathrm e}^{\frac {3}{{\mathrm e}^{\frac {19}{5}}-22}} \left (1-{\mathrm e}^{\frac {x \left ({\mathrm e}^{\frac {19}{5}}-27\right )}{{\mathrm e}^{\frac {19}{5}}-22}}\right )}{{\mathrm e}^{\frac {19}{5}}-27}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.41, size = 17, normalized size = 0.89 \begin {gather*} e^{\left (\frac {x e^{\frac {19}{5}} - 27 \, x + 3}{e^{\frac {19}{5}} - 22}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 0.11, size = 31, normalized size = 1.63 \begin {gather*} {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.13, size = 19, normalized size = 1.00 \begin {gather*} e^{\frac {- 27 x + x e^{\frac {19}{5}} + 3}{-22 + e^{\frac {19}{5}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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