3.83.62 \(\int \frac {e^{6+2 x} (-2+4 x)}{5 e^{12+4 x}-10 e^{6+2 x} x+5 x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac {2 x}{5 \left (-e^{2 (3+x)}+x\right )} \]

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Rubi [A]  time = 0.31, antiderivative size = 20, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6688, 12, 6711, 32} \begin {gather*} \frac {2}{5 \left (1-\frac {e^{2 x+6}}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(6 + 2*x)*(-2 + 4*x))/(5*E^(12 + 4*x) - 10*E^(6 + 2*x)*x + 5*x^2),x]

[Out]

2/(5*(1 - E^(6 + 2*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 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 6688

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

Rule 6711

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w
, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}
, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{6+2 x} (-1+2 x)}{5 \left (e^{6+2 x}-x\right )^2} \, dx\\ &=\frac {2}{5} \int \frac {e^{6+2 x} (-1+2 x)}{\left (e^{6+2 x}-x\right )^2} \, dx\\ &=\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^2} \, dx,x,\frac {e^{6+2 x}}{x}\right )\\ &=\frac {2}{5 \left (1-\frac {e^{6+2 x}}{x}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2 x}{5 \left (e^{6+2 x}-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(6 + 2*x)*(-2 + 4*x))/(5*E^(12 + 4*x) - 10*E^(6 + 2*x)*x + 5*x^2),x]

[Out]

(-2*x)/(5*(E^(6 + 2*x) - x))

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fricas [A]  time = 0.60, size = 15, normalized size = 0.83 \begin {gather*} \frac {2 \, x}{5 \, {\left (x - e^{\left (2 \, x + 6\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-2)*exp(2*x+6)/(5*exp(2*x+6)^2-10*x*exp(2*x+6)+5*x^2),x, algorithm="fricas")

[Out]

2/5*x/(x - e^(2*x + 6))

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giac [A]  time = 0.21, size = 15, normalized size = 0.83 \begin {gather*} \frac {2 \, x}{5 \, {\left (x - e^{\left (2 \, x + 6\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-2)*exp(2*x+6)/(5*exp(2*x+6)^2-10*x*exp(2*x+6)+5*x^2),x, algorithm="giac")

[Out]

2/5*x/(x - e^(2*x + 6))

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maple [A]  time = 0.11, size = 16, normalized size = 0.89

method	result	size
risch	\(\frac {2 x}{5 \left (x -{\mathrm e}^{2 x +6}\right )}\)	\(16\)
norman	\(\frac {2 \,{\mathrm e}^{2 x +6}}{5 \left (x -{\mathrm e}^{2 x +6}\right )}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x-2)*exp(2*x+6)/(5*exp(2*x+6)^2-10*x*exp(2*x+6)+5*x^2),x,method=_RETURNVERBOSE)

[Out]

2/5*x/(x-exp(2*x+6))

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maxima [A]  time = 0.40, size = 15, normalized size = 0.83 \begin {gather*} \frac {2 \, x}{5 \, {\left (x - e^{\left (2 \, x + 6\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-2)*exp(2*x+6)/(5*exp(2*x+6)^2-10*x*exp(2*x+6)+5*x^2),x, algorithm="maxima")

[Out]

2/5*x/(x - e^(2*x + 6))

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mupad [B]  time = 0.08, size = 17, normalized size = 0.94 \begin {gather*} \frac {2\,x}{5\,x-5\,{\mathrm {e}}^{2\,x+6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x + 6)*(4*x - 2))/(5*exp(4*x + 12) - 10*x*exp(2*x + 6) + 5*x^2),x)

[Out]

(2*x)/(5*x - 5*exp(2*x + 6))

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sympy [A]  time = 0.10, size = 15, normalized size = 0.83 \begin {gather*} - \frac {2 x}{- 5 x + 5 e^{2 x + 6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-2)*exp(2*x+6)/(5*exp(2*x+6)**2-10*x*exp(2*x+6)+5*x**2),x)

[Out]

-2*x/(-5*x + 5*exp(2*x + 6))

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