3.13.90 \(\int \frac {-264+192 x}{256+16 e^6-352 x+249 x^2-88 x^3+16 x^4+e^3 (-128+88 x-32 x^2)} \, dx\)

Optimal. Leaf size=19 \[ \frac {6}{-4+e^3+\frac {11 x}{4}-x^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {1680, 12, 261} \begin {gather*} -\frac {384}{64 \left (x-\frac {11}{8}\right )^2-64 e^3+135} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-264 + 192*x)/(256 + 16*E^6 - 352*x + 249*x^2 - 88*x^3 + 16*x^4 + E^3*(-128 + 88*x - 32*x^2)),x]

[Out]

-384/(135 - 64*E^3 + 64*(-11/8 + x)^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {49152 x}{\left (135-64 e^3+64 x^2\right )^2} \, dx,x,-\frac {11}{8}+x\right )\\ &=49152 \operatorname {Subst}\left (\int \frac {x}{\left (135-64 e^3+64 x^2\right )^2} \, dx,x,-\frac {11}{8}+x\right )\\ &=-\frac {384}{135-64 e^3+(-11+8 x)^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} -\frac {24}{16-4 e^3-11 x+4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-264 + 192*x)/(256 + 16*E^6 - 352*x + 249*x^2 - 88*x^3 + 16*x^4 + E^3*(-128 + 88*x - 32*x^2)),x]

[Out]

-24/(16 - 4*E^3 - 11*x + 4*x^2)

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fricas [A]  time = 1.13, size = 18, normalized size = 0.95 \begin {gather*} -\frac {24}{4 \, x^{2} - 11 \, x - 4 \, e^{3} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((192*x-264)/(16*exp(3)^2+(-32*x^2+88*x-128)*exp(3)+16*x^4-88*x^3+249*x^2-352*x+256),x, algorithm="fr
icas")

[Out]

-24/(4*x^2 - 11*x - 4*e^3 + 16)

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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((192*x-264)/(16*exp(3)^2+(-32*x^2+88*x-128)*exp(3)+16*x^4-88*x^3+249*x^2-352*x+256),x, algorithm="gi
ac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 6/sqrt(-exp(3)^2+exp(6))*atan((8*sageVAR
x^2/2-11*sageVARx-4*exp(3)+16)/4/sqrt(-exp(3)^2+exp(6)))

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maple [A]  time = 0.07, size = 17, normalized size = 0.89

method	result	size
risch	\(\frac {6}{\frac {11 x}{4}-x^{2}-4+{\mathrm e}^{3}}\)	\(17\)
gosper	\(\frac {24}{-4 x^{2}+4 \,{\mathrm e}^{3}+11 x -16}\)	\(19\)
norman	\(\frac {24}{-4 x^{2}+4 \,{\mathrm e}^{3}+11 x -16}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((192*x-264)/(16*exp(3)^2+(-32*x^2+88*x-128)*exp(3)+16*x^4-88*x^3+249*x^2-352*x+256),x,method=_RETURNVERBOS
E)

[Out]

6/(11/4*x-x^2-4+exp(3))

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maxima [A]  time = 0.34, size = 18, normalized size = 0.95 \begin {gather*} -\frac {24}{4 \, x^{2} - 11 \, x - 4 \, e^{3} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((192*x-264)/(16*exp(3)^2+(-32*x^2+88*x-128)*exp(3)+16*x^4-88*x^3+249*x^2-352*x+256),x, algorithm="ma
xima")

[Out]

-24/(4*x^2 - 11*x - 4*e^3 + 16)

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mupad [B]  time = 0.17, size = 18, normalized size = 0.95 \begin {gather*} \frac {24}{-4\,x^2+11\,x+4\,{\mathrm {e}}^3-16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((192*x - 264)/(16*exp(6) - 352*x - exp(3)*(32*x^2 - 88*x + 128) + 249*x^2 - 88*x^3 + 16*x^4 + 256),x)

[Out]

24/(11*x + 4*exp(3) - 4*x^2 - 16)

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sympy [A]  time = 0.27, size = 17, normalized size = 0.89 \begin {gather*} - \frac {24}{4 x^{2} - 11 x - 4 e^{3} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((192*x-264)/(16*exp(3)**2+(-32*x**2+88*x-128)*exp(3)+16*x**4-88*x**3+249*x**2-352*x+256),x)

[Out]

-24/(4*x**2 - 11*x - 4*exp(3) + 16)

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