3.68.12 \(\int \frac {-72-2 e^{e^2}+48 x-12 x^2+x^3}{-64+48 x-12 x^2+x^3} \, dx\)

Optimal. Leaf size=24 \[ e^3+x+\frac {4+e^{e^2}}{-16 x+(4+x)^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 0.71, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2074} \begin {gather*} x+\frac {4+e^{e^2}}{(4-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-72 - 2*E^E^2 + 48*x - 12*x^2 + x^3)/(-64 + 48*x - 12*x^2 + x^3),x]

[Out]

(4 + E^E^2)/(4 - x)^2 + x

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {2 \left (4+e^{e^2}\right )}{(-4+x)^3}\right ) \, dx\\ &=\frac {4+e^{e^2}}{(4-x)^2}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 0.62 \begin {gather*} \frac {4+e^{e^2}}{(-4+x)^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-72 - 2*E^E^2 + 48*x - 12*x^2 + x^3)/(-64 + 48*x - 12*x^2 + x^3),x]

[Out]

(4 + E^E^2)/(-4 + x)^2 + x

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fricas [A]  time = 0.60, size = 27, normalized size = 1.12 \begin {gather*} \frac {x^{3} - 8 \, x^{2} + 16 \, x + e^{\left (e^{2}\right )} + 4}{x^{2} - 8 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(exp(2))+x^3-12*x^2+48*x-72)/(x^3-12*x^2+48*x-64),x, algorithm="fricas")

[Out]

(x^3 - 8*x^2 + 16*x + e^(e^2) + 4)/(x^2 - 8*x + 16)

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giac [A]  time = 0.18, size = 13, normalized size = 0.54 \begin {gather*} x + \frac {e^{\left (e^{2}\right )} + 4}{{\left (x - 4\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(exp(2))+x^3-12*x^2+48*x-72)/(x^3-12*x^2+48*x-64),x, algorithm="giac")

[Out]

x + (e^(e^2) + 4)/(x - 4)^2

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maple [A]  time = 0.04, size = 17, normalized size = 0.71

method	result	size
default	\(x -\frac {-8-2 \,{\mathrm e}^{{\mathrm e}^{2}}}{2 \left (x -4\right )^{2}}\)	\(17\)
norman	\(\frac {x^{3}+{\mathrm e}^{{\mathrm e}^{2}}-48 x +132}{\left (x -4\right )^{2}}\)	\(18\)
gosper	\(\frac {x^{3}+{\mathrm e}^{{\mathrm e}^{2}}-48 x +132}{x^{2}-8 x +16}\)	\(23\)
risch	\(x +\frac {{\mathrm e}^{{\mathrm e}^{2}}}{x^{2}-8 x +16}+\frac {4}{x^{2}-8 x +16}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*exp(exp(2))+x^3-12*x^2+48*x-72)/(x^3-12*x^2+48*x-64),x,method=_RETURNVERBOSE)

[Out]

x-1/2*(-8-2*exp(exp(2)))/(x-4)^2

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maxima [A]  time = 0.36, size = 18, normalized size = 0.75 \begin {gather*} x + \frac {e^{\left (e^{2}\right )} + 4}{x^{2} - 8 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(exp(2))+x^3-12*x^2+48*x-72)/(x^3-12*x^2+48*x-64),x, algorithm="maxima")

[Out]

x + (e^(e^2) + 4)/(x^2 - 8*x + 16)

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mupad [B]  time = 0.06, size = 13, normalized size = 0.54 \begin {gather*} x+\frac {{\mathrm {e}}^{{\mathrm {e}}^2}+4}{{\left (x-4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*exp(exp(2)) - 48*x + 12*x^2 - x^3 + 72)/(48*x - 12*x^2 + x^3 - 64),x)

[Out]

x + (exp(exp(2)) + 4)/(x - 4)^2

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sympy [A]  time = 0.16, size = 15, normalized size = 0.62 \begin {gather*} x + \frac {4 + e^{e^{2}}}{x^{2} - 8 x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(exp(2))+x**3-12*x**2+48*x-72)/(x**3-12*x**2+48*x-64),x)

[Out]

x + (4 + exp(exp(2)))/(x**2 - 8*x + 16)

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