3.901 \(\int \frac{-2+7 x}{42-16 x+2 x^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{7}{4} \log \left (x^2-8 x+21\right )-\frac{13 \tan ^{-1}\left (\frac{4-x}{\sqrt{5}}\right )}{\sqrt{5}} \]

[Out]

(-13*ArcTan[(4 - x)/Sqrt[5]])/Sqrt[5] + (7*Log[21 - 8*x + x^2])/4

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Rubi [A]  time = 0.0221802, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {634, 618, 204, 628} \[ \frac{7}{4} \log \left (x^2-8 x+21\right )-\frac{13 \tan ^{-1}\left (\frac{4-x}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[(-2 + 7*x)/(42 - 16*x + 2*x^2),x]

[Out]

(-13*ArcTan[(4 - x)/Sqrt[5]])/Sqrt[5] + (7*Log[21 - 8*x + x^2])/4

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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]

Rubi steps

\begin{align*} \int \frac{-2+7 x}{42-16 x+2 x^2} \, dx &=\frac{7}{4} \int \frac{-16+4 x}{42-16 x+2 x^2} \, dx+26 \int \frac{1}{42-16 x+2 x^2} \, dx\\ &=\frac{7}{4} \log \left (21-8 x+x^2\right )-52 \operatorname{Subst}\left (\int \frac{1}{-80-x^2} \, dx,x,-16+4 x\right )\\ &=-\frac{13 \tan ^{-1}\left (\frac{4-x}{\sqrt{5}}\right )}{\sqrt{5}}+\frac{7}{4} \log \left (21-8 x+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0125885, size = 35, normalized size = 1.06 \[ \frac{1}{2} \left (\frac{7}{2} \log \left (x^2-8 x+21\right )+\frac{26 \tan ^{-1}\left (\frac{x-4}{\sqrt{5}}\right )}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 7*x)/(42 - 16*x + 2*x^2),x]

[Out]

((26*ArcTan[(-4 + x)/Sqrt[5]])/Sqrt[5] + (7*Log[21 - 8*x + x^2])/2)/2

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Maple [A]  time = 0.003, size = 29, normalized size = 0.9 \begin{align*}{\frac{7\,\ln \left ({x}^{2}-8\,x+21 \right ) }{4}}+{\frac{13\,\sqrt{5}}{5}\arctan \left ({\frac{ \left ( 2\,x-8 \right ) \sqrt{5}}{10}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2+7*x)/(2*x^2-16*x+42),x)

[Out]

7/4*ln(x^2-8*x+21)+13/5*5^(1/2)*arctan(1/10*(2*x-8)*5^(1/2))

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Maxima [A]  time = 1.52334, size = 35, normalized size = 1.06 \begin{align*} \frac{13}{5} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (x - 4\right )}\right ) + \frac{7}{4} \, \log \left (x^{2} - 8 \, x + 21\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+7*x)/(2*x^2-16*x+42),x, algorithm="maxima")

[Out]

13/5*sqrt(5)*arctan(1/5*sqrt(5)*(x - 4)) + 7/4*log(x^2 - 8*x + 21)

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Fricas [A]  time = 1.52322, size = 92, normalized size = 2.79 \begin{align*} \frac{13}{5} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (x - 4\right )}\right ) + \frac{7}{4} \, \log \left (x^{2} - 8 \, x + 21\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+7*x)/(2*x^2-16*x+42),x, algorithm="fricas")

[Out]

13/5*sqrt(5)*arctan(1/5*sqrt(5)*(x - 4)) + 7/4*log(x^2 - 8*x + 21)

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Sympy [A]  time = 0.135162, size = 39, normalized size = 1.18 \begin{align*} \frac{7 \log{\left (x^{2} - 8 x + 21 \right )}}{4} + \frac{13 \sqrt{5} \operatorname{atan}{\left (\frac{\sqrt{5} x}{5} - \frac{4 \sqrt{5}}{5} \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+7*x)/(2*x**2-16*x+42),x)

[Out]

7*log(x**2 - 8*x + 21)/4 + 13*sqrt(5)*atan(sqrt(5)*x/5 - 4*sqrt(5)/5)/5

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Giac [A]  time = 1.22752, size = 35, normalized size = 1.06 \begin{align*} \frac{13}{5} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (x - 4\right )}\right ) + \frac{7}{4} \, \log \left (x^{2} - 8 \, x + 21\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+7*x)/(2*x^2-16*x+42),x, algorithm="giac")

[Out]

13/5*sqrt(5)*arctan(1/5*sqrt(5)*(x - 4)) + 7/4*log(x^2 - 8*x + 21)