3.99.19 \(\int \frac {-274-68 x-4 x^2}{289+68 x+4 x^2} \, dx\)

Optimal. Leaf size=32 \[ 3-x-\frac {x}{x+\frac {x+x^2}{16+x}}+\log (i \pi +\log (5)) \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 0.47, 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*} -x-\frac {15}{2 (2 x+17)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-274 - 68*x - 4*x^2)/(289 + 68*x + 4*x^2),x]

[Out]

-x - 15/(2*(17 + 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 {-274-68 x-4 x^2}{(17+2 x)^2} \, dx\\ &=\int \left (-1+\frac {15}{(17+2 x)^2}\right ) \, dx\\ &=-x-\frac {15}{2 (17+2 x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.53 \begin {gather*} -\frac {2 \left (76+17 x+x^2\right )}{17+2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-274 - 68*x - 4*x^2)/(289 + 68*x + 4*x^2),x]

[Out]

(-2*(76 + 17*x + x^2))/(17 + 2*x)

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fricas [A]  time = 0.63, size = 19, normalized size = 0.59 \begin {gather*} -\frac {4 \, x^{2} + 34 \, x + 15}{2 \, {\left (2 \, x + 17\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-68*x-274)/(4*x^2+68*x+289),x, algorithm="fricas")

[Out]

-1/2*(4*x^2 + 34*x + 15)/(2*x + 17)

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giac [A]  time = 0.15, size = 13, normalized size = 0.41 \begin {gather*} -x - \frac {15}{2 \, {\left (2 \, x + 17\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-68*x-274)/(4*x^2+68*x+289),x, algorithm="giac")

[Out]

-x - 15/2/(2*x + 17)

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maple [A]  time = 0.11, size = 12, normalized size = 0.38

method	result	size
risch	\(-x -\frac {15}{4 \left (x +\frac {17}{2}\right )}\)	\(12\)
default	\(-x -\frac {15}{2 \left (2 x +17\right )}\)	\(14\)
gosper	\(-\frac {2 x^{2}-137}{2 x +17}\)	\(17\)
meijerg	\(\frac {304 x}{289 \left (1+\frac {2 x}{17}\right )}-\frac {x \left (\frac {6 x}{17}+6\right )}{3 \left (1+\frac {2 x}{17}\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2-68*x-274)/(4*x^2+68*x+289),x,method=_RETURNVERBOSE)

[Out]

-x-15/4/(x+17/2)

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maxima [A]  time = 0.37, size = 13, normalized size = 0.41 \begin {gather*} -x - \frac {15}{2 \, {\left (2 \, x + 17\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-68*x-274)/(4*x^2+68*x+289),x, algorithm="maxima")

[Out]

-x - 15/2/(2*x + 17)

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mupad [B]  time = 0.04, size = 13, normalized size = 0.41 \begin {gather*} -x-\frac {15}{4\,\left (x+\frac {17}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(68*x + 4*x^2 + 274)/(68*x + 4*x^2 + 289),x)

[Out]

- x - 15/(4*(x + 17/2))

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sympy [A]  time = 0.07, size = 8, normalized size = 0.25 \begin {gather*} - x - \frac {15}{4 x + 34} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2-68*x-274)/(4*x**2+68*x+289),x)

[Out]

-x - 15/(4*x + 34)

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