3.84.94 \(\int \frac {-32+32 x-8 x^2+e^{e^4} (-400 x+100 x^2)}{4-4 x+x^2} \, dx\)

Optimal. Leaf size=22 \[ 4 \left (5-2 x+\frac {25 e^{e^4} x^2}{-2+x}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.23, number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {27, 1850} \begin {gather*} -4 \left (2-25 e^{e^4}\right ) x-\frac {400 e^{e^4}}{2-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-32 + 32*x - 8*x^2 + E^E^4*(-400*x + 100*x^2))/(4 - 4*x + x^2),x]

[Out]

(-400*E^E^4)/(2 - x) - 4*(2 - 25*E^E^4)*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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-32+32 x-8 x^2+e^{e^4} \left (-400 x+100 x^2\right )}{(-2+x)^2} \, dx\\ &=\int \left (4 \left (-2+25 e^{e^4}\right )-\frac {400 e^{e^4}}{(-2+x)^2}\right ) \, dx\\ &=-\frac {400 e^{e^4}}{2-x}-4 \left (2-25 e^{e^4}\right ) x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 28, normalized size = 1.27 \begin {gather*} 4 \left (\frac {100 e^{e^4}}{-2+x}+\left (-2+25 e^{e^4}\right ) (-2+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32 + 32*x - 8*x^2 + E^E^4*(-400*x + 100*x^2))/(4 - 4*x + x^2),x]

[Out]

4*((100*E^E^4)/(-2 + x) + (-2 + 25*E^E^4)*(-2 + x))

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fricas [A]  time = 0.55, size = 29, normalized size = 1.32 \begin {gather*} -\frac {4 \, {\left (2 \, x^{2} - 25 \, {\left (x^{2} - 2 \, x + 4\right )} e^{\left (e^{4}\right )} - 4 \, x\right )}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^2-400*x)*exp(exp(4))-8*x^2+32*x-32)/(x^2-4*x+4),x, algorithm="fricas")

[Out]

-4*(2*x^2 - 25*(x^2 - 2*x + 4)*e^(e^4) - 4*x)/(x - 2)

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giac [A]  time = 0.14, size = 20, normalized size = 0.91 \begin {gather*} 100 \, x e^{\left (e^{4}\right )} - 8 \, x + \frac {400 \, e^{\left (e^{4}\right )}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^2-400*x)*exp(exp(4))-8*x^2+32*x-32)/(x^2-4*x+4),x, algorithm="giac")

[Out]

100*x*e^(e^4) - 8*x + 400*e^(e^4)/(x - 2)

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maple [A]  time = 0.43, size = 20, normalized size = 0.91

method	result	size
norman	\(\frac {\left (100 \,{\mathrm e}^{{\mathrm e}^{4}}-8\right ) x^{2}+32}{x -2}\)	\(20\)
default	\(100 x \,{\mathrm e}^{{\mathrm e}^{4}}-8 x +\frac {400 \,{\mathrm e}^{{\mathrm e}^{4}}}{x -2}\)	\(21\)
risch	\(100 x \,{\mathrm e}^{{\mathrm e}^{4}}-8 x +\frac {400 \,{\mathrm e}^{{\mathrm e}^{4}}}{x -2}\)	\(21\)
gosper	\(\frac {100 x^{2} {\mathrm e}^{{\mathrm e}^{4}}-8 x^{2}+32}{x -2}\)	\(23\)
meijerg	\(-\frac {8 x}{1-\frac {x}{2}}-2 \left (100 \,{\mathrm e}^{{\mathrm e}^{4}}-8\right ) \left (-\frac {x \left (-\frac {3 x}{2}+6\right )}{6 \left (1-\frac {x}{2}\right )}-2 \ln \left (1-\frac {x}{2}\right )\right )-2 \left (200 \,{\mathrm e}^{{\mathrm e}^{4}}-16\right ) \left (\frac {x}{2-x}+\ln \left (1-\frac {x}{2}\right )\right )\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((100*x^2-400*x)*exp(exp(4))-8*x^2+32*x-32)/(x^2-4*x+4),x,method=_RETURNVERBOSE)

[Out]

((100*exp(exp(4))-8)*x^2+32)/(x-2)

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maxima [A]  time = 0.36, size = 21, normalized size = 0.95 \begin {gather*} 4 \, x {\left (25 \, e^{\left (e^{4}\right )} - 2\right )} + \frac {400 \, e^{\left (e^{4}\right )}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^2-400*x)*exp(exp(4))-8*x^2+32*x-32)/(x^2-4*x+4),x, algorithm="maxima")

[Out]

4*x*(25*e^(e^4) - 2) + 400*e^(e^4)/(x - 2)

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mupad [B]  time = 5.14, size = 20, normalized size = 0.91 \begin {gather*} x\,\left (100\,{\mathrm {e}}^{{\mathrm {e}}^4}-8\right )+\frac {400\,{\mathrm {e}}^{{\mathrm {e}}^4}}{x-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(4))*(400*x - 100*x^2) - 32*x + 8*x^2 + 32)/(x^2 - 4*x + 4),x)

[Out]

x*(100*exp(exp(4)) - 8) + (400*exp(exp(4)))/(x - 2)

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sympy [A]  time = 0.12, size = 19, normalized size = 0.86 \begin {gather*} - x \left (8 - 100 e^{e^{4}}\right ) + \frac {400 e^{e^{4}}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x**2-400*x)*exp(exp(4))-8*x**2+32*x-32)/(x**2-4*x+4),x)

[Out]

-x*(8 - 100*exp(exp(4))) + 400*exp(exp(4))/(x - 2)

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