3.23.45 \(\int \frac {-96+176 x-104 x^2}{16+4 x+13 x^2} \, dx\)

Optimal. Leaf size=17 \[ 8 \left (-x+\log \left (4+x+\frac {13 x^2}{4}\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {1657, 628} \begin {gather*} 8 \log \left (13 x^2+4 x+16\right )-8 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-96 + 176*x - 104*x^2)/(16 + 4*x + 13*x^2),x]

[Out]

-8*x + 8*Log[16 + 4*x + 13*x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-8+\frac {16 (2+13 x)}{16+4 x+13 x^2}\right ) \, dx\\ &=-8 x+16 \int \frac {2+13 x}{16+4 x+13 x^2} \, dx\\ &=-8 x+8 \log \left (16+4 x+13 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -8 \left (x-\log \left (16+4 x+13 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-96 + 176*x - 104*x^2)/(16 + 4*x + 13*x^2),x]

[Out]

-8*(x - Log[16 + 4*x + 13*x^2])

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fricas [A]  time = 0.71, size = 17, normalized size = 1.00 \begin {gather*} -8 \, x + 8 \, \log \left (13 \, x^{2} + 4 \, x + 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-104*x^2+176*x-96)/(13*x^2+4*x+16),x, algorithm="fricas")

[Out]

-8*x + 8*log(13*x^2 + 4*x + 16)

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giac [A]  time = 0.17, size = 17, normalized size = 1.00 \begin {gather*} -8 \, x + 8 \, \log \left (13 \, x^{2} + 4 \, x + 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-104*x^2+176*x-96)/(13*x^2+4*x+16),x, algorithm="giac")

[Out]

-8*x + 8*log(13*x^2 + 4*x + 16)

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maple [A]  time = 0.81, size = 18, normalized size = 1.06

method	result	size
default	\(-8 x +8 \ln \left (13 x^{2}+4 x +16\right )\)	\(18\)
norman	\(-8 x +8 \ln \left (13 x^{2}+4 x +16\right )\)	\(18\)
risch	\(-8 x +8 \ln \left (13 x^{2}+4 x +16\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-104*x^2+176*x-96)/(13*x^2+4*x+16),x,method=_RETURNVERBOSE)

[Out]

-8*x+8*ln(13*x^2+4*x+16)

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maxima [A]  time = 0.46, size = 17, normalized size = 1.00 \begin {gather*} -8 \, x + 8 \, \log \left (13 \, x^{2} + 4 \, x + 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-104*x^2+176*x-96)/(13*x^2+4*x+16),x, algorithm="maxima")

[Out]

-8*x + 8*log(13*x^2 + 4*x + 16)

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mupad [B]  time = 0.05, size = 17, normalized size = 1.00 \begin {gather*} 8\,\ln \left (13\,x^2+4\,x+16\right )-8\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(104*x^2 - 176*x + 96)/(4*x + 13*x^2 + 16),x)

[Out]

8*log(4*x + 13*x^2 + 16) - 8*x

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sympy [A]  time = 0.08, size = 15, normalized size = 0.88 \begin {gather*} - 8 x + 8 \log {\left (13 x^{2} + 4 x + 16 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-104*x**2+176*x-96)/(13*x**2+4*x+16),x)

[Out]

-8*x + 8*log(13*x**2 + 4*x + 16)

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