3.83.25 \(\int \frac {e^{\frac {-30+e (-25+4 x)}{-6+e (-5+2 x)}} (36 e+30 e^2)}{36+e (60-24 x)+e^2 (25-20 x+4 x^2)} \, dx\)

Optimal. Leaf size=23 \[ e^{5+\frac {x}{\frac {1}{e}+\frac {1}{3} \left (\frac {5}{2}-x\right )}} \]

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Rubi [A]  time = 0.27, antiderivative size = 22, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {12, 6688, 2230, 2209} \begin {gather*} e^{\frac {3 (6+5 e)}{-2 e x+5 e+6}+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-30 + E*(-25 + 4*x))/(-6 + E*(-5 + 2*x)))*(36*E + 30*E^2))/(36 + E*(60 - 24*x) + E^2*(25 - 20*x + 4*x
^2)),x]

[Out]

E^(2 + (3*(6 + 5*E))/(6 + 5*E - 2*E*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 2230

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :>
Int[(g + h*x)^m*F^((d*e + b*f)/d - (f*(b*c - a*d))/(d*(c + d*x))), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m},
x] && NeQ[b*c - a*d, 0] && EqQ[d*g - c*h, 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} &=(6 e (6+5 e)) \int \frac {\exp \left (\frac {-30+e (-25+4 x)}{-6+e (-5+2 x)}\right )}{36+e (60-24 x)+e^2 \left (25-20 x+4 x^2\right )} \, dx\\ &=(6 e (6+5 e)) \int \frac {e^{\frac {30+25 e-4 e x}{6+5 e-2 e x}}}{(6+5 e-2 e x)^2} \, dx\\ &=(6 e (6+5 e)) \int \frac {\exp \left (2+\frac {-4 e (6+5 e)+2 e (30+25 e)}{2 e (6+5 e-2 e x)}\right )}{(6+5 e-2 e x)^2} \, dx\\ &=e^{2+\frac {3 (6+5 e)}{6+5 e-2 e x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 23, normalized size = 1.00 \begin {gather*} e^{\frac {30+25 e-4 e x}{6+5 e-2 e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-30 + E*(-25 + 4*x))/(-6 + E*(-5 + 2*x)))*(36*E + 30*E^2))/(36 + E*(60 - 24*x) + E^2*(25 - 20*x
+ 4*x^2)),x]

[Out]

E^((30 + 25*E - 4*E*x)/(6 + 5*E - 2*E*x))

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fricas [A]  time = 1.17, size = 24, normalized size = 1.04 \begin {gather*} e^{\left (\frac {{\left (4 \, x - 25\right )} e - 30}{{\left (2 \, x - 5\right )} e - 6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*exp(1)^2+36*exp(1))*exp(((4*x-25)*exp(1)-30)/((2*x-5)*exp(1)-6))/((4*x^2-20*x+25)*exp(1)^2+(-24*
x+60)*exp(1)+36),x, algorithm="fricas")

[Out]

e^(((4*x - 25)*e - 30)/((2*x - 5)*e - 6))

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giac [B]  time = 1.50, size = 46, normalized size = 2.00 \begin {gather*} \frac {{\left (5 \, e^{2} + 6 \, e\right )} e^{\left (\frac {4 \, x e - 25 \, e - 30}{2 \, x e - 5 \, e - 6} - 1\right )}}{5 \, e + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*exp(1)^2+36*exp(1))*exp(((4*x-25)*exp(1)-30)/((2*x-5)*exp(1)-6))/((4*x^2-20*x+25)*exp(1)^2+(-24*
x+60)*exp(1)+36),x, algorithm="giac")

[Out]

(5*e^2 + 6*e)*e^((4*x*e - 25*e - 30)/(2*x*e - 5*e - 6) - 1)/(5*e + 6)

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maple [A]  time = 0.42, size = 27, normalized size = 1.17

method	result	size
gosper	\({\mathrm e}^{\frac {4 x \,{\mathrm e}-25 \,{\mathrm e}-30}{2 x \,{\mathrm e}-5 \,{\mathrm e}-6}}\)	\(27\)
derivativedivides	\(-\frac {\left (5 \,{\mathrm e}^{2}+6 \,{\mathrm e}\right ) {\mathrm e}^{-1} \left (-15 \,{\mathrm e}-18\right ) {\mathrm e}^{2+\frac {-15 \,{\mathrm e}-18}{2 x \,{\mathrm e}-5 \,{\mathrm e}-6}}}{3 \left (25 \,{\mathrm e}^{2}+60 \,{\mathrm e}+36\right )}\)	\(61\)
default	\(-\frac {\left (30 \,{\mathrm e}^{2}+36 \,{\mathrm e}\right ) {\mathrm e}^{-1} \left (-15 \,{\mathrm e}-18\right ) {\mathrm e}^{2+\frac {-15 \,{\mathrm e}-18}{2 x \,{\mathrm e}-5 \,{\mathrm e}-6}}}{18 \left (25 \,{\mathrm e}^{2}+60 \,{\mathrm e}+36\right )}\)	\(61\)
norman	\(\frac {\left (-5 \,{\mathrm e}-6\right ) {\mathrm e}^{\frac {\left (4 x -25\right ) {\mathrm e}-30}{\left (2 x -5\right ) {\mathrm e}-6}}+2 x \,{\mathrm e} \,{\mathrm e}^{\frac {\left (4 x -25\right ) {\mathrm e}-30}{\left (2 x -5\right ) {\mathrm e}-6}}}{2 x \,{\mathrm e}-5 \,{\mathrm e}-6}\)	\(76\)
risch	\(\frac {5 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}-20 \,{\mathrm e}-24}{2 x \,{\mathrm e}-5 \,{\mathrm e}-6}} {\mathrm e}^{2}}{5 \,{\mathrm e}+6}+\frac {6 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}-20 \,{\mathrm e}-24}{2 x \,{\mathrm e}-5 \,{\mathrm e}-6}} {\mathrm e}}{5 \,{\mathrm e}+6}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((30*exp(1)^2+36*exp(1))*exp(((4*x-25)*exp(1)-30)/((2*x-5)*exp(1)-6))/((4*x^2-20*x+25)*exp(1)^2+(-24*x+60)*
exp(1)+36),x,method=_RETURNVERBOSE)

[Out]

exp((4*x*exp(1)-25*exp(1)-30)/(2*x*exp(1)-5*exp(1)-6))

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maxima [B]  time = 0.39, size = 53, normalized size = 2.30 \begin {gather*} \frac {{\left (5 \, e^{2} + 6 \, e\right )} e^{\left (-\frac {15 \, e}{2 \, x e - 5 \, e - 6} - \frac {18}{2 \, x e - 5 \, e - 6} + 1\right )}}{5 \, e + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*exp(1)^2+36*exp(1))*exp(((4*x-25)*exp(1)-30)/((2*x-5)*exp(1)-6))/((4*x^2-20*x+25)*exp(1)^2+(-24*
x+60)*exp(1)+36),x, algorithm="maxima")

[Out]

(5*e^2 + 6*e)*e^(-15*e/(2*x*e - 5*e - 6) - 18/(2*x*e - 5*e - 6) + 1)/(5*e + 6)

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mupad [B]  time = 7.77, size = 54, normalized size = 2.35 \begin {gather*} {\mathrm {e}}^{-\frac {4\,x\,\mathrm {e}}{5\,\mathrm {e}-2\,x\,\mathrm {e}+6}}\,{\mathrm {e}}^{\frac {30}{5\,\mathrm {e}-2\,x\,\mathrm {e}+6}}\,{\mathrm {e}}^{\frac {25\,\mathrm {e}}{5\,\mathrm {e}-2\,x\,\mathrm {e}+6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((exp(1)*(4*x - 25) - 30)/(exp(1)*(2*x - 5) - 6))*(36*exp(1) + 30*exp(2)))/(exp(2)*(4*x^2 - 20*x + 25)
- exp(1)*(24*x - 60) + 36),x)

[Out]

exp(-(4*x*exp(1))/(5*exp(1) - 2*x*exp(1) + 6))*exp(30/(5*exp(1) - 2*x*exp(1) + 6))*exp((25*exp(1))/(5*exp(1) -
2*x*exp(1) + 6))

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sympy [A]  time = 0.35, size = 20, normalized size = 0.87 \begin {gather*} e^{\frac {e \left (4 x - 25\right ) - 30}{e \left (2 x - 5\right ) - 6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*exp(1)**2+36*exp(1))*exp(((4*x-25)*exp(1)-30)/((2*x-5)*exp(1)-6))/((4*x**2-20*x+25)*exp(1)**2+(-
24*x+60)*exp(1)+36),x)

[Out]

exp((E*(4*x - 25) - 30)/(E*(2*x - 5) - 6))

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