3.100.4 \(\int \frac {43+12 x-6 x^2}{6-12 x+6 x^2} \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 0.47, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {27, 12, 683} \begin {gather*} \frac {49}{6 (1-x)}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(43 + 12*x - 6*x^2)/(6 - 12*x + 6*x^2),x]

[Out]

49/(6*(1 - x)) - x

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 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {43+12 x-6 x^2}{6 (-1+x)^2} \, dx\\ &=\frac {1}{6} \int \frac {43+12 x-6 x^2}{(-1+x)^2} \, dx\\ &=\frac {1}{6} \int \left (-6+\frac {49}{(-1+x)^2}\right ) \, dx\\ &=\frac {49}{6 (1-x)}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.53 \begin {gather*} \frac {1}{6} \left (-\frac {49}{-1+x}-6 (-1+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(43 + 12*x - 6*x^2)/(6 - 12*x + 6*x^2),x]

[Out]

(-49/(-1 + x) - 6*(-1 + x))/6

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fricas [A]  time = 0.62, size = 17, normalized size = 0.53 \begin {gather*} -\frac {6 \, x^{2} - 6 \, x + 49}{6 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2+12*x+43)/(6*x^2-12*x+6),x, algorithm="fricas")

[Out]

-1/6*(6*x^2 - 6*x + 49)/(x - 1)

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giac [A]  time = 0.15, size = 11, normalized size = 0.34 \begin {gather*} -x - \frac {49}{6 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2+12*x+43)/(6*x^2-12*x+6),x, algorithm="giac")

[Out]

-x - 49/6/(x - 1)

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maple [A]  time = 0.14, size = 12, normalized size = 0.38

method	result	size
default	\(-x -\frac {49}{6 \left (x -1\right )}\)	\(12\)
risch	\(-x -\frac {49}{6 \left (x -1\right )}\)	\(12\)
gosper	\(-\frac {6 x^{2}+43}{6 \left (x -1\right )}\)	\(15\)
meijerg	\(\frac {55 x}{6 \left (1-x \right )}-\frac {x \left (-3 x +6\right )}{3 \left (1-x \right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^2+12*x+43)/(6*x^2-12*x+6),x,method=_RETURNVERBOSE)

[Out]

-x-49/6/(x-1)

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maxima [A]  time = 0.35, size = 11, normalized size = 0.34 \begin {gather*} -x - \frac {49}{6 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2+12*x+43)/(6*x^2-12*x+6),x, algorithm="maxima")

[Out]

-x - 49/6/(x - 1)

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mupad [B]  time = 0.04, size = 13, normalized size = 0.41 \begin {gather*} -x-\frac {49}{6\,\left (x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x - 6*x^2 + 43)/(6*x^2 - 12*x + 6),x)

[Out]

- x - 49/(6*(x - 1))

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sympy [A]  time = 0.07, size = 8, normalized size = 0.25 \begin {gather*} - x - \frac {49}{6 x - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**2+12*x+43)/(6*x**2-12*x+6),x)

[Out]

-x - 49/(6*x - 6)

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