3.35.22 \(\int \frac {19+2 x-2 x^2}{-17+2 x+2 x^2} \, dx\)

Optimal. Leaf size=15 \[ -x+\log \left (9-2 \left (-4+x+x^2\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 15, 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*} \log \left (-2 x^2-2 x+17\right )-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(19 + 2*x - 2*x^2)/(-17 + 2*x + 2*x^2),x]

[Out]

-x + Log[17 - 2*x - 2*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 (-1+\frac {2 (1+2 x)}{-17+2 x+2 x^2}\right ) \, dx\\ &=-x+2 \int \frac {1+2 x}{-17+2 x+2 x^2} \, dx\\ &=-x+\log \left (17-2 x-2 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -x+\log \left (17-2 x-2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(19 + 2*x - 2*x^2)/(-17 + 2*x + 2*x^2),x]

[Out]

-x + Log[17 - 2*x - 2*x^2]

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fricas [A]  time = 0.52, size = 15, normalized size = 1.00 \begin {gather*} -x + \log \left (2 \, x^{2} + 2 \, x - 17\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+2*x+19)/(2*x^2+2*x-17),x, algorithm="fricas")

[Out]

-x + log(2*x^2 + 2*x - 17)

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giac [A]  time = 0.13, size = 16, normalized size = 1.07 \begin {gather*} -x + \log \left ({\left | 2 \, x^{2} + 2 \, x - 17 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+2*x+19)/(2*x^2+2*x-17),x, algorithm="giac")

[Out]

-x + log(abs(2*x^2 + 2*x - 17))

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maple [A]  time = 0.55, size = 16, normalized size = 1.07

method	result	size
default	\(-x +\ln \left (2 x^{2}+2 x -17\right )\)	\(16\)
norman	\(-x +\ln \left (2 x^{2}+2 x -17\right )\)	\(16\)
risch	\(-x +\ln \left (2 x^{2}+2 x -17\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2+2*x+19)/(2*x^2+2*x-17),x,method=_RETURNVERBOSE)

[Out]

-x+ln(2*x^2+2*x-17)

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maxima [A]  time = 0.47, size = 15, normalized size = 1.00 \begin {gather*} -x + \log \left (2 \, x^{2} + 2 \, x - 17\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+2*x+19)/(2*x^2+2*x-17),x, algorithm="maxima")

[Out]

-x + log(2*x^2 + 2*x - 17)

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mupad [B]  time = 0.05, size = 11, normalized size = 0.73 \begin {gather*} \ln \left (x^2+x-\frac {17}{2}\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 2*x^2 + 19)/(2*x + 2*x^2 - 17),x)

[Out]

log(x + x^2 - 17/2) - x

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sympy [A]  time = 0.08, size = 12, normalized size = 0.80 \begin {gather*} - x + \log {\left (2 x^{2} + 2 x - 17 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2+2*x+19)/(2*x**2+2*x-17),x)

[Out]

-x + log(2*x**2 + 2*x - 17)

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