3.101.37 \(\int \frac {9-24 x}{e^3 (9 x^2-24 x^3+16 x^4)} \, dx\)

Optimal. Leaf size=15 \[ \frac {3}{e^3 x (-3+4 x)} \]

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {12, 1594, 27, 74} \begin {gather*} -\frac {3}{e^3 (3-4 x) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9 - 24*x)/(E^3*(9*x^2 - 24*x^3 + 16*x^4)),x]

[Out]

-3/(E^3*(3 - 4*x)*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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {9-24 x}{9 x^2-24 x^3+16 x^4} \, dx}{e^3}\\ &=\frac {\int \frac {9-24 x}{x^2 \left (9-24 x+16 x^2\right )} \, dx}{e^3}\\ &=\frac {\int \frac {9-24 x}{x^2 (-3+4 x)^2} \, dx}{e^3}\\ &=-\frac {3}{e^3 (3-4 x) x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {3}{e^3 x (-3+4 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 - 24*x)/(E^3*(9*x^2 - 24*x^3 + 16*x^4)),x]

[Out]

3/(E^3*x*(-3 + 4*x))

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fricas [A]  time = 1.35, size = 15, normalized size = 1.00 \begin {gather*} \frac {3 \, e^{\left (-3\right )}}{4 \, x^{2} - 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*e^(-3)/(4*x^2 - 3*x)

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giac [A]  time = 0.18, size = 15, normalized size = 1.00 \begin {gather*} \frac {3 \, e^{\left (-3\right )}}{4 \, x^{2} - 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*e^(-3)/(4*x^2 - 3*x)

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maple [A]  time = 0.04, size = 15, normalized size = 1.00

method	result	size
risch	\(\frac {3 \,{\mathrm e}^{-3}}{x \left (4 x -3\right )}\)	\(15\)
gosper	\(\frac {3 \,{\mathrm e}^{-3}}{x \left (4 x -3\right )}\)	\(17\)
norman	\(\frac {3 \,{\mathrm e}^{-3}}{x \left (4 x -3\right )}\)	\(17\)
default	\(3 \,{\mathrm e}^{-3} \left (\frac {4}{3 \left (4 x -3\right )}-\frac {1}{3 x}\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/x*exp(-3)/(4*x-3)

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maxima [A]  time = 0.38, size = 15, normalized size = 1.00 \begin {gather*} \frac {3 \, e^{\left (-3\right )}}{4 \, x^{2} - 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*e^(-3)/(4*x^2 - 3*x)

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mupad [B]  time = 7.42, size = 14, normalized size = 0.93 \begin {gather*} \frac {3\,{\mathrm {e}}^{-3}}{x\,\left (4\,x-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-3)*(24*x - 9))/(9*x^2 - 24*x^3 + 16*x^4),x)

[Out]

(3*exp(-3))/(x*(4*x - 3))

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sympy [A]  time = 0.18, size = 15, normalized size = 1.00 \begin {gather*} \frac {3}{4 x^{2} e^{3} - 3 x e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x+9)/(16*x**4-24*x**3+9*x**2)/exp(3),x)

[Out]

3/(4*x**2*exp(3) - 3*x*exp(3))

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