3.5.75 \(\int \frac {-5-4 x-4 x^2}{1+4 x+4 x^2} \, dx\)

Optimal. Leaf size=19 \[ 2-x+\log \left (e^{-4+\frac {2}{1+2 x}}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.68, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {27, 683} \begin {gather*} \frac {2}{2 x+1}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 - 4*x - 4*x^2)/(1 + 4*x + 4*x^2),x]

[Out]

-x + 2/(1 + 2*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 {-5-4 x-4 x^2}{(1+2 x)^2} \, dx\\ &=\int \left (-1-\frac {4}{(1+2 x)^2}\right ) \, dx\\ &=-x+\frac {2}{1+2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.68 \begin {gather*} -x+\frac {2}{1+2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 - 4*x - 4*x^2)/(1 + 4*x + 4*x^2),x]

[Out]

-x + 2/(1 + 2*x)

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fricas [A]  time = 0.54, size = 17, normalized size = 0.89 \begin {gather*} -\frac {2 \, x^{2} + x - 2}{2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-4*x-5)/(4*x^2+4*x+1),x, algorithm="fricas")

[Out]

-(2*x^2 + x - 2)/(2*x + 1)

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giac [A]  time = 0.35, size = 13, normalized size = 0.68 \begin {gather*} -x + \frac {2}{2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-4*x-5)/(4*x^2+4*x+1),x, algorithm="giac")

[Out]

-x + 2/(2*x + 1)

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maple [A]  time = 0.30, size = 10, normalized size = 0.53

method	result	size
risch	\(-x +\frac {1}{\frac {1}{2}+x}\)	\(10\)
default	\(-x +\frac {2}{2 x +1}\)	\(14\)
gosper	\(-\frac {x \left (5+2 x \right )}{2 x +1}\)	\(16\)
meijerg	\(-\frac {x \left (6 x +6\right )}{3 \left (2 x +1\right )}-\frac {3 x}{2 x +1}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2-4*x-5)/(4*x^2+4*x+1),x,method=_RETURNVERBOSE)

[Out]

-x+1/(1/2+x)

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maxima [A]  time = 0.62, size = 13, normalized size = 0.68 \begin {gather*} -x + \frac {2}{2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-4*x-5)/(4*x^2+4*x+1),x, algorithm="maxima")

[Out]

-x + 2/(2*x + 1)

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mupad [B]  time = 0.04, size = 9, normalized size = 0.47 \begin {gather*} \frac {1}{x+\frac {1}{2}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 4*x^2 + 5)/(4*x + 4*x^2 + 1),x)

[Out]

1/(x + 1/2) - x

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sympy [A]  time = 0.07, size = 7, normalized size = 0.37 \begin {gather*} - x + \frac {2}{2 x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2-4*x-5)/(4*x**2+4*x+1),x)

[Out]

-x + 2/(2*x + 1)

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