3.9.1 \(\int \frac {-8192+e^6+6144 x-1536 x^2+128 x^3}{-8192+6144 x-1536 x^2+128 x^3} \, dx\)

Optimal. Leaf size=17 \[ -4-\frac {e^6}{256 (4-x)^2}+x \]

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Rubi [A]  time = 0.04, antiderivative size = 16, normalized size of antiderivative = 0.94, 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 {e^6}{256 (4-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8192 + E^6 + 6144*x - 1536*x^2 + 128*x^3)/(-8192 + 6144*x - 1536*x^2 + 128*x^3),x]

[Out]

-1/256*E^6/(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 {e^6}{128 (-4+x)^3}\right ) \, dx\\ &=-\frac {e^6}{256 (4-x)^2}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.29 \begin {gather*} \frac {1}{128} \left (-\frac {e^6}{2 (-4+x)^2}+128 (-4+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8192 + E^6 + 6144*x - 1536*x^2 + 128*x^3)/(-8192 + 6144*x - 1536*x^2 + 128*x^3),x]

[Out]

(-1/2*E^6/(-4 + x)^2 + 128*(-4 + x))/128

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fricas [B]  time = 0.60, size = 30, normalized size = 1.76 \begin {gather*} \frac {256 \, x^{3} - 2048 \, x^{2} + 4096 \, x - e^{6}}{256 \, {\left (x^{2} - 8 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3)^2+128*x^3-1536*x^2+6144*x-8192)/(128*x^3-1536*x^2+6144*x-8192),x, algorithm="fricas")

[Out]

1/256*(256*x^3 - 2048*x^2 + 4096*x - e^6)/(x^2 - 8*x + 16)

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giac [A]  time = 0.39, size = 11, normalized size = 0.65 \begin {gather*} x - \frac {e^{6}}{256 \, {\left (x - 4\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3)^2+128*x^3-1536*x^2+6144*x-8192)/(128*x^3-1536*x^2+6144*x-8192),x, algorithm="giac")

[Out]

x - 1/256*e^6/(x - 4)^2

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maple [A]  time = 0.04, size = 12, normalized size = 0.71

method	result	size
default	\(x -\frac {{\mathrm e}^{6}}{256 \left (x -4\right )^{2}}\)	\(12\)
risch	\(x -\frac {{\mathrm e}^{6}}{256 \left (x^{2}-8 x +16\right )}\)	\(17\)
norman	\(\frac {x^{3}-48 x +128-\frac {{\mathrm e}^{6}}{256}}{\left (x -4\right )^{2}}\)	\(21\)
gosper	\(-\frac {-256 x^{3}+{\mathrm e}^{6}+12288 x -32768}{256 \left (x^{2}-8 x +16\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3)^2+128*x^3-1536*x^2+6144*x-8192)/(128*x^3-1536*x^2+6144*x-8192),x,method=_RETURNVERBOSE)

[Out]

x-1/256*exp(6)/(x-4)^2

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maxima [A]  time = 0.56, size = 16, normalized size = 0.94 \begin {gather*} x - \frac {e^{6}}{256 \, {\left (x^{2} - 8 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3)^2+128*x^3-1536*x^2+6144*x-8192)/(128*x^3-1536*x^2+6144*x-8192),x, algorithm="maxima")

[Out]

x - 1/256*e^6/(x^2 - 8*x + 16)

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mupad [B]  time = 0.66, size = 11, normalized size = 0.65 \begin {gather*} x-\frac {{\mathrm {e}}^6}{256\,{\left (x-4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6144*x + exp(6) - 1536*x^2 + 128*x^3 - 8192)/(6144*x - 1536*x^2 + 128*x^3 - 8192),x)

[Out]

x - exp(6)/(256*(x - 4)^2)

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sympy [A]  time = 0.11, size = 14, normalized size = 0.82 \begin {gather*} x - \frac {e^{6}}{256 x^{2} - 2048 x + 4096} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3)**2+128*x**3-1536*x**2+6144*x-8192)/(128*x**3-1536*x**2+6144*x-8192),x)

[Out]

x - exp(6)/(256*x**2 - 2048*x + 4096)

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