3.35.64 \(\int \frac {e^{-\frac {5 x}{-3+4 e^{8-2 x} x^4}} (15+e^{8-2 x} (60 x^4-40 x^5))}{9-24 e^{8-2 x} x^4+16 e^{16-4 x} x^8} \, dx\)

Optimal. Leaf size=21 \[ e^{\frac {5 x}{3-4 e^{8-2 x} x^4}} \]

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Rubi [A]  time = 0.91, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6706} \begin {gather*} e^{\frac {5 x}{3-4 e^{8-2 x} x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(15 + E^(8 - 2*x)*(60*x^4 - 40*x^5))/(E^((5*x)/(-3 + 4*E^(8 - 2*x)*x^4))*(9 - 24*E^(8 - 2*x)*x^4 + 16*E^(1
6 - 4*x)*x^8)),x]

[Out]

E^((5*x)/(3 - 4*E^(8 - 2*x)*x^4))

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{\frac {5 x}{3-4 e^{8-2 x} x^4}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.24, size = 21, normalized size = 1.00 \begin {gather*} e^{-\frac {5 x}{-3+4 e^{8-2 x} x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15 + E^(8 - 2*x)*(60*x^4 - 40*x^5))/(E^((5*x)/(-3 + 4*E^(8 - 2*x)*x^4))*(9 - 24*E^(8 - 2*x)*x^4 + 1
6*E^(16 - 4*x)*x^8)),x]

[Out]

E^((-5*x)/(-3 + 4*E^(8 - 2*x)*x^4))

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fricas [A]  time = 1.10, size = 19, normalized size = 0.90 \begin {gather*} e^{\left (-\frac {5 \, x}{4 \, x^{4} e^{\left (-2 \, x + 8\right )} - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^5+60*x^4)*exp(-x+4)^2+15)*exp(-5*x/(4*x^4*exp(-x+4)^2-3))/(16*x^8*exp(-x+4)^4-24*x^4*exp(-x+
4)^2+9),x, algorithm="fricas")

[Out]

e^(-5*x/(4*x^4*e^(-2*x + 8) - 3))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^5+60*x^4)*exp(-x+4)^2+15)*exp(-5*x/(4*x^4*exp(-x+4)^2-3))/(16*x^8*exp(-x+4)^4-24*x^4*exp(-x+
4)^2+9),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.11, size = 20, normalized size = 0.95

method	result	size
risch	\({\mathrm e}^{-\frac {5 x}{4 x^{4} {\mathrm e}^{-2 x +8}-3}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-40*x^5+60*x^4)*exp(-x+4)^2+15)*exp(-5*x/(4*x^4*exp(-x+4)^2-3))/(16*x^8*exp(-x+4)^4-24*x^4*exp(-x+4)^2+9
),x,method=_RETURNVERBOSE)

[Out]

exp(-5*x/(4*x^4*exp(-2*x+8)-3))

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maxima [A]  time = 0.66, size = 24, normalized size = 1.14 \begin {gather*} e^{\left (-\frac {5 \, x e^{\left (2 \, x\right )}}{4 \, x^{4} e^{8} - 3 \, e^{\left (2 \, x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^5+60*x^4)*exp(-x+4)^2+15)*exp(-5*x/(4*x^4*exp(-x+4)^2-3))/(16*x^8*exp(-x+4)^4-24*x^4*exp(-x+
4)^2+9),x, algorithm="maxima")

[Out]

e^(-5*x*e^(2*x)/(4*x^4*e^8 - 3*e^(2*x)))

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mupad [B]  time = 2.18, size = 19, normalized size = 0.90 \begin {gather*} {\mathrm {e}}^{-\frac {5\,x}{4\,x^4\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^8-3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(5*x)/(4*x^4*exp(8 - 2*x) - 3))*(exp(8 - 2*x)*(60*x^4 - 40*x^5) + 15))/(16*x^8*exp(16 - 4*x) - 24*x^
4*exp(8 - 2*x) + 9),x)

[Out]

exp(-(5*x)/(4*x^4*exp(-2*x)*exp(8) - 3))

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sympy [A]  time = 0.35, size = 19, normalized size = 0.90 \begin {gather*} e^{- \frac {5 x}{4 x^{4} e^{8 - 2 x} - 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x**5+60*x**4)*exp(-x+4)**2+15)*exp(-5*x/(4*x**4*exp(-x+4)**2-3))/(16*x**8*exp(-x+4)**4-24*x**4
*exp(-x+4)**2+9),x)

[Out]

exp(-5*x/(4*x**4*exp(8 - 2*x) - 3))

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