3.55.32 \(\int -\frac {6075 e^{\frac {225}{-19+3 e+3 x}}}{361+9 e^2-114 x+9 x^2+e (-114+18 x)} \, dx\)

Optimal. Leaf size=20 \[ 9 \left (4+e^{\frac {225}{5+3 e+3 (-8+x)}}\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 6688, 2209} \begin {gather*} 9 e^{-\frac {225}{-3 x-3 e+19}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6075*E^(225/(-19 + 3*E + 3*x)))/(361 + 9*E^2 - 114*x + 9*x^2 + E*(-114 + 18*x)),x]

[Out]

9/E^(225/(19 - 3*E - 3*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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (6075 \int \frac {e^{\frac {225}{-19+3 e+3 x}}}{361+9 e^2-114 x+9 x^2+e (-114+18 x)} \, dx\right )\\ &=-\left (6075 \int \frac {e^{\frac {225}{-19+3 e+3 x}}}{(19-3 e-3 x)^2} \, dx\right )\\ &=9 e^{-\frac {225}{19-3 e-3 x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.80 \begin {gather*} 9 e^{\frac {225}{-19+3 e+3 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6075*E^(225/(-19 + 3*E + 3*x)))/(361 + 9*E^2 - 114*x + 9*x^2 + E*(-114 + 18*x)),x]

[Out]

9*E^(225/(-19 + 3*E + 3*x))

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fricas [A]  time = 1.76, size = 16, normalized size = 0.80 \begin {gather*} 9 \, e^{\left (\frac {225}{3 \, x + 3 \, e - 19}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-6075*exp(225/(3*exp(1)+3*x-19))/(9*exp(1)^2+(18*x-114)*exp(1)+9*x^2-114*x+361),x, algorithm="fricas
")

[Out]

9*e^(225/(3*x + 3*e - 19))

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giac [A]  time = 0.26, size = 16, normalized size = 0.80 \begin {gather*} 9 \, e^{\left (\frac {225}{3 \, x + 3 \, e - 19}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-6075*exp(225/(3*exp(1)+3*x-19))/(9*exp(1)^2+(18*x-114)*exp(1)+9*x^2-114*x+361),x, algorithm="giac")

[Out]

9*e^(225/(3*x + 3*e - 19))

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maple [A]  time = 0.30, size = 17, normalized size = 0.85

method	result	size
gosper	\(9 \,{\mathrm e}^{\frac {225}{3 \,{\mathrm e}+3 x -19}}\)	\(17\)
derivativedivides	\(9 \,{\mathrm e}^{\frac {225}{3 \,{\mathrm e}+3 x -19}}\)	\(17\)
default	\(9 \,{\mathrm e}^{\frac {225}{3 \,{\mathrm e}+3 x -19}}\)	\(17\)
risch	\(9 \,{\mathrm e}^{\frac {225}{3 \,{\mathrm e}+3 x -19}}\)	\(17\)
norman	\(\frac {\left (-171+27 \,{\mathrm e}\right ) {\mathrm e}^{\frac {225}{3 \,{\mathrm e}+3 x -19}}+27 x \,{\mathrm e}^{\frac {225}{3 \,{\mathrm e}+3 x -19}}}{3 \,{\mathrm e}+3 x -19}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-6075*exp(225/(3*exp(1)+3*x-19))/(9*exp(1)^2+(18*x-114)*exp(1)+9*x^2-114*x+361),x,method=_RETURNVERBOSE)

[Out]

9*exp(225/(3*exp(1)+3*x-19))

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maxima [A]  time = 0.37, size = 16, normalized size = 0.80 \begin {gather*} 9 \, e^{\left (\frac {225}{3 \, x + 3 \, e - 19}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-6075*exp(225/(3*exp(1)+3*x-19))/(9*exp(1)^2+(18*x-114)*exp(1)+9*x^2-114*x+361),x, algorithm="maxima
")

[Out]

9*e^(225/(3*x + 3*e - 19))

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mupad [B]  time = 0.75, size = 16, normalized size = 0.80 \begin {gather*} 9\,{\mathrm {e}}^{\frac {225}{3\,x+3\,\mathrm {e}-19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6075*exp(225/(3*x + 3*exp(1) - 19)))/(9*exp(2) - 114*x + 9*x^2 + exp(1)*(18*x - 114) + 361),x)

[Out]

9*exp(225/(3*x + 3*exp(1) - 19))

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sympy [A]  time = 0.25, size = 14, normalized size = 0.70 \begin {gather*} 9 e^{\frac {225}{3 x - 19 + 3 e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-6075*exp(225/(3*exp(1)+3*x-19))/(9*exp(1)**2+(18*x-114)*exp(1)+9*x**2-114*x+361),x)

[Out]

9*exp(225/(3*x - 19 + 3*E))

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