3.74.53 \(\int \frac {75 e^{4+x}}{625 e^2+81 e^{2 x}+450 e^{1+x}} \, dx\)

Optimal. Leaf size=19 \[ \frac {e^3}{3+\frac {25 e^{1-x}}{3}} \]

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Rubi [A]  time = 0.05, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 2282, 32} \begin {gather*} -\frac {25 e^4}{3 \left (9 e^x+25 e\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(75*E^(4 + x))/(625*E^2 + 81*E^(2*x) + 450*E^(1 + x)),x]

[Out]

(-25*E^4)/(3*(25*E + 9*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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=75 \int \frac {e^{4+x}}{625 e^2+81 e^{2 x}+450 e^{1+x}} \, dx\\ &=75 \operatorname {Subst}\left (\int \frac {e^4}{(25 e+9 x)^2} \, dx,x,e^x\right )\\ &=\left (75 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{(25 e+9 x)^2} \, dx,x,e^x\right )\\ &=-\frac {25 e^4}{3 \left (25 e+9 e^x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 0.95 \begin {gather*} -\frac {25 e^4}{3 \left (25 e+9 e^x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(75*E^(4 + x))/(625*E^2 + 81*E^(2*x) + 450*E^(1 + x)),x]

[Out]

(-25*E^4)/(3*(25*E + 9*E^x))

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fricas [A]  time = 0.87, size = 17, normalized size = 0.89 \begin {gather*} -\frac {25 \, e^{8}}{3 \, {\left (25 \, e^{5} + 9 \, e^{\left (x + 4\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(75*exp(1)*exp(3)*exp(x)/(81*exp(x)^2+450*exp(1)*exp(x)+625*exp(1)^2),x, algorithm="fricas")

[Out]

-25/3*e^8/(25*e^5 + 9*e^(x + 4))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(75*exp(1)*exp(3)*exp(x)/(81*exp(x)^2+450*exp(1)*exp(x)+625*exp(1)^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 75*exp(4)/450/sqrt(exp(1)^2-exp(2))*ln(s
qrt((162*exp(sageVARx)+450*exp(1))^2+(-450*sqrt(-exp(1)^2+exp(2)))^2)/sqrt((162*exp(sageVARx)+450*exp(1))^2+(4
50*sqrt(-exp(1)^2+exp

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maple [A]  time = 0.06, size = 16, normalized size = 0.84

method	result	size
risch	\(-\frac {25 \,{\mathrm e}^{4}}{3 \left (25 \,{\mathrm e}+9 \,{\mathrm e}^{x}\right )}\)	\(16\)
norman	\(-\frac {25 \,{\mathrm e} \,{\mathrm e}^{3}}{3 \left (25 \,{\mathrm e}+9 \,{\mathrm e}^{x}\right )}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(75*exp(1)*exp(3)*exp(x)/(81*exp(x)^2+450*exp(1)*exp(x)+625*exp(1)^2),x,method=_RETURNVERBOSE)

[Out]

-25/3*exp(4)/(25*exp(1)+9*exp(x))

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maxima [A]  time = 0.36, size = 15, normalized size = 0.79 \begin {gather*} -\frac {25 \, e^{4}}{3 \, {\left (25 \, e + 9 \, e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(75*exp(1)*exp(3)*exp(x)/(81*exp(x)^2+450*exp(1)*exp(x)+625*exp(1)^2),x, algorithm="maxima")

[Out]

-25/3*e^4/(25*e + 9*e^x)

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mupad [B]  time = 0.11, size = 17, normalized size = 0.89 \begin {gather*} -\frac {25\,{\mathrm {e}}^5}{3\,\left (9\,{\mathrm {e}}^{x+1}+25\,{\mathrm {e}}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((75*exp(4)*exp(x))/(81*exp(2*x) + 625*exp(2) + 450*exp(1)*exp(x)),x)

[Out]

-(25*exp(5))/(3*(9*exp(x + 1) + 25*exp(2)))

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sympy [A]  time = 0.09, size = 15, normalized size = 0.79 \begin {gather*} - \frac {25 e^{4}}{27 e^{x} + 75 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(75*exp(1)*exp(3)*exp(x)/(81*exp(x)**2+450*exp(1)*exp(x)+625*exp(1)**2),x)

[Out]

-25*exp(4)/(27*exp(x) + 75*E)

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