3.71.58 \(\int \frac {12-40 x+58 x^2-24 x^3+e^4 (-30+120 x-120 x^2)}{e^4 (3-12 x+12 x^2)} \, dx\)

Optimal. Leaf size=32 \[ 2 (3+x) \left (-5+\frac {3+\frac {1}{3} \left (-x+\frac {x^2}{1-2 x}\right )}{e^4}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {12, 27, 1850} \begin {gather*} -\frac {x^2}{e^4}+\frac {\left (17-60 e^4\right ) x}{6 e^4}+\frac {7}{12 e^4 (1-2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12 - 40*x + 58*x^2 - 24*x^3 + E^4*(-30 + 120*x - 120*x^2))/(E^4*(3 - 12*x + 12*x^2)),x]

[Out]

7/(12*E^4*(1 - 2*x)) + ((17 - 60*E^4)*x)/(6*E^4) - x^2/E^4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {\int \frac {12-40 x+58 x^2-24 x^3+e^4 \left (-30+120 x-120 x^2\right )}{3-12 x+12 x^2} \, dx}{e^4}\\ &=\frac {\int \frac {12-40 x+58 x^2-24 x^3+e^4 \left (-30+120 x-120 x^2\right )}{3 (-1+2 x)^2} \, dx}{e^4}\\ &=\frac {\int \frac {12-40 x+58 x^2-24 x^3+e^4 \left (-30+120 x-120 x^2\right )}{(-1+2 x)^2} \, dx}{3 e^4}\\ &=\frac {\int \left (\frac {1}{2} \left (17-60 e^4\right )-6 x+\frac {7}{2 (-1+2 x)^2}\right ) \, dx}{3 e^4}\\ &=\frac {7}{12 e^4 (1-2 x)}+\frac {\left (17-60 e^4\right ) x}{6 e^4}-\frac {x^2}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 41, normalized size = 1.28 \begin {gather*} -\frac {2 \left (-7+60 e^4 (1-2 x)^2+62 x-80 x^2+24 x^3\right )}{3 e^4 (-8+16 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12 - 40*x + 58*x^2 - 24*x^3 + E^4*(-30 + 120*x - 120*x^2))/(E^4*(3 - 12*x + 12*x^2)),x]

[Out]

(-2*(-7 + 60*E^4*(1 - 2*x)^2 + 62*x - 80*x^2 + 24*x^3))/(3*E^4*(-8 + 16*x))

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fricas [A]  time = 0.72, size = 39, normalized size = 1.22 \begin {gather*} -\frac {{\left (24 \, x^{3} - 80 \, x^{2} + 120 \, {\left (2 \, x^{2} - x\right )} e^{4} + 34 \, x + 7\right )} e^{\left (-4\right )}}{12 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-120*x^2+120*x-30)*exp(4)-24*x^3+58*x^2-40*x+12)/(12*x^2-12*x+3)/exp(4),x, algorithm="fricas")

[Out]

-1/12*(24*x^3 - 80*x^2 + 120*(2*x^2 - x)*e^4 + 34*x + 7)*e^(-4)/(2*x - 1)

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giac [A]  time = 0.14, size = 27, normalized size = 0.84 \begin {gather*} -\frac {1}{12} \, {\left (12 \, x^{2} + 120 \, x e^{4} - 34 \, x + \frac {7}{2 \, x - 1}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-120*x^2+120*x-30)*exp(4)-24*x^3+58*x^2-40*x+12)/(12*x^2-12*x+3)/exp(4),x, algorithm="giac")

[Out]

-1/12*(12*x^2 + 120*x*e^4 - 34*x + 7/(2*x - 1))*e^(-4)

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maple [A]  time = 0.14, size = 26, normalized size = 0.81

method	result	size
risch	\(-10 x -x^{2} {\mathrm e}^{-4}+\frac {17 x \,{\mathrm e}^{-4}}{6}-\frac {7 \,{\mathrm e}^{-4}}{24 \left (x -\frac {1}{2}\right )}\)	\(26\)
default	\(\frac {{\mathrm e}^{-4} \left (-3 x^{2}+\frac {17 x}{2}-30 x \,{\mathrm e}^{4}-\frac {7}{4 \left (2 x -1\right )}\right )}{3}\)	\(30\)
gosper	\(-\frac {\left (60 x^{2} {\mathrm e}^{4}+6 x^{3}-20 x^{2}-15 \,{\mathrm e}^{4}+6\right ) {\mathrm e}^{-4}}{3 \left (2 x -1\right )}\)	\(37\)
norman	\(\frac {-2 \,{\mathrm e}^{-4} x^{3}-\frac {20 \left (-1+3 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4} x^{2}}{3}+\left (5 \,{\mathrm e}^{4}-2\right ) {\mathrm e}^{-4}}{2 x -1}\)	\(45\)
meijerg	\(\frac {4 \,{\mathrm e}^{-4} x}{1-2 x}-\frac {\left (-120 \,{\mathrm e}^{4}+58\right ) {\mathrm e}^{-4} \left (-\frac {2 x \left (6-6 x \right )}{3 \left (1-2 x \right )}-2 \ln \left (1-2 x \right )\right )}{24}+\frac {\left (120 \,{\mathrm e}^{4}-40\right ) {\mathrm e}^{-4} \left (\frac {2 x}{1-2 x}+\ln \left (1-2 x \right )\right )}{12}-\frac {{\mathrm e}^{-4} \left (\frac {x \left (-8 x^{2}-12 x +12\right )}{-4 x +2}+3 \ln \left (1-2 x \right )\right )}{2}-\frac {10 x}{1-2 x}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-120*x^2+120*x-30)*exp(4)-24*x^3+58*x^2-40*x+12)/(12*x^2-12*x+3)/exp(4),x,method=_RETURNVERBOSE)

[Out]

-10*x-x^2*exp(-4)+17/6*x*exp(-4)-7/24*exp(-4)/(x-1/2)

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maxima [A]  time = 0.38, size = 28, normalized size = 0.88 \begin {gather*} -\frac {1}{12} \, {\left (12 \, x^{2} + 2 \, x {\left (60 \, e^{4} - 17\right )} + \frac {7}{2 \, x - 1}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-120*x^2+120*x-30)*exp(4)-24*x^3+58*x^2-40*x+12)/(12*x^2-12*x+3)/exp(4),x, algorithm="maxima")

[Out]

-1/12*(12*x^2 + 2*x*(60*e^4 - 17) + 7/(2*x - 1))*e^(-4)

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mupad [B]  time = 4.15, size = 40, normalized size = 1.25 \begin {gather*} \frac {7}{2\,\left (6\,{\mathrm {e}}^4-12\,x\,{\mathrm {e}}^4\right )}-x^2\,{\mathrm {e}}^{-4}-x\,\left (2\,{\mathrm {e}}^{-4}+\frac {{\mathrm {e}}^{-4}\,\left (120\,{\mathrm {e}}^4-58\right )}{12}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4)*(40*x + exp(4)*(120*x^2 - 120*x + 30) - 58*x^2 + 24*x^3 - 12))/(12*x^2 - 12*x + 3),x)

[Out]

7/(2*(6*exp(4) - 12*x*exp(4))) - x^2*exp(-4) - x*(2*exp(-4) + (exp(-4)*(120*exp(4) - 58))/12)

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sympy [A]  time = 0.21, size = 31, normalized size = 0.97 \begin {gather*} - \frac {x^{2}}{e^{4}} - x \left (10 - \frac {17}{6 e^{4}}\right ) - \frac {7}{24 x e^{4} - 12 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-120*x**2+120*x-30)*exp(4)-24*x**3+58*x**2-40*x+12)/(12*x**2-12*x+3)/exp(4),x)

[Out]

-x**2*exp(-4) - x*(10 - 17*exp(-4)/6) - 7/(24*x*exp(4) - 12*exp(4))

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