3.169 \(\int \frac {5-4 x}{(-2-4 x+3 x^2)^2} \, dx\)

Optimal. Leaf size=43 \[ -\frac {18-7 x}{20 \left (-3 x^2+4 x+2\right )}-\frac {7 \tanh ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{20 \sqrt {10}} \]

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {638, 618, 206} \[ -\frac {18-7 x}{20 \left (-3 x^2+4 x+2\right )}-\frac {7 \tanh ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{20 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - 4*x)/(-2 - 4*x + 3*x^2)^2,x]

[Out]

-(18 - 7*x)/(20*(2 + 4*x - 3*x^2)) - (7*ArcTanh[(2 - 3*x)/Sqrt[10]])/(20*Sqrt[10])

Rule 206

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

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 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps

\begin {align*} \int \frac {5-4 x}{\left (-2-4 x+3 x^2\right )^2} \, dx &=-\frac {18-7 x}{20 \left (2+4 x-3 x^2\right )}-\frac {7}{20} \int \frac {1}{-2-4 x+3 x^2} \, dx\\ &=-\frac {18-7 x}{20 \left (2+4 x-3 x^2\right )}+\frac {7}{10} \operatorname {Subst}\left (\int \frac {1}{40-x^2} \, dx,x,-4+6 x\right )\\ &=-\frac {18-7 x}{20 \left (2+4 x-3 x^2\right )}-\frac {7 \tanh ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{20 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 62, normalized size = 1.44 \[ \frac {18-7 x}{20 \left (3 x^2-4 x-2\right )}-\frac {7 \log \left (-3 x+\sqrt {10}+2\right )}{40 \sqrt {10}}+\frac {7 \log \left (3 x+\sqrt {10}-2\right )}{40 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - 4*x)/(-2 - 4*x + 3*x^2)^2,x]

[Out]

(18 - 7*x)/(20*(-2 - 4*x + 3*x^2)) - (7*Log[2 + Sqrt[10] - 3*x])/(40*Sqrt[10]) + (7*Log[-2 + Sqrt[10] + 3*x])/
(40*Sqrt[10])

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IntegrateAlgebraic [A]  time = 0.06, size = 48, normalized size = 1.12 \[ \frac {18-7 x}{20 \left (3 x^2-4 x-2\right )}-\frac {7 \tanh ^{-1}\left (\sqrt {\frac {2}{5}}-\frac {3 x}{\sqrt {10}}\right )}{20 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(5 - 4*x)/(-2 - 4*x + 3*x^2)^2,x]

[Out]

(18 - 7*x)/(20*(-2 - 4*x + 3*x^2)) - (7*ArcTanh[Sqrt[2/5] - (3*x)/Sqrt[10]])/(20*Sqrt[10])

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fricas [A]  time = 1.05, size = 68, normalized size = 1.58 \[ \frac {7 \, \sqrt {10} {\left (3 \, x^{2} - 4 \, x - 2\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {10} {\left (3 \, x - 2\right )} - 12 \, x + 14}{3 \, x^{2} - 4 \, x - 2}\right ) - 140 \, x + 360}{400 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-4*x)/(3*x^2-4*x-2)^2,x, algorithm="fricas")

[Out]

1/400*(7*sqrt(10)*(3*x^2 - 4*x - 2)*log((9*x^2 + 2*sqrt(10)*(3*x - 2) - 12*x + 14)/(3*x^2 - 4*x - 2)) - 140*x
+ 360)/(3*x^2 - 4*x - 2)

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giac [A]  time = 0.95, size = 51, normalized size = 1.19 \[ -\frac {7}{400} \, \sqrt {10} \log \left (\frac {{\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}}\right ) - \frac {7 \, x - 18}{20 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-4*x)/(3*x^2-4*x-2)^2,x, algorithm="giac")

[Out]

-7/400*sqrt(10)*log(abs(6*x - 2*sqrt(10) - 4)/abs(6*x + 2*sqrt(10) - 4)) - 1/20*(7*x - 18)/(3*x^2 - 4*x - 2)

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maple [A]  time = 0.35, size = 37, normalized size = 0.86

method	result	size
default	\(-\frac {14 x -36}{40 \left (3 x^{2}-4 x -2\right )}+\frac {7 \sqrt {10}\, \arctanh \left (\frac {\left (6 x -4\right ) \sqrt {10}}{20}\right )}{200}\)	\(37\)
risch	\(\frac {-\frac {7 x}{60}+\frac {3}{10}}{x^{2}-\frac {4}{3} x -\frac {2}{3}}+\frac {7 \sqrt {10}\, \ln \left (3 x -2+\sqrt {10}\right )}{400}-\frac {7 \sqrt {10}\, \ln \left (3 x -2-\sqrt {10}\right )}{400}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-4*x)/(3*x^2-4*x-2)^2,x,method=_RETURNVERBOSE)

[Out]

-1/40*(14*x-36)/(3*x^2-4*x-2)+7/200*10^(1/2)*arctanh(1/20*(6*x-4)*10^(1/2))

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maxima [A]  time = 0.95, size = 47, normalized size = 1.09 \[ -\frac {7}{400} \, \sqrt {10} \log \left (\frac {3 \, x - \sqrt {10} - 2}{3 \, x + \sqrt {10} - 2}\right ) - \frac {7 \, x - 18}{20 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-4*x)/(3*x^2-4*x-2)^2,x, algorithm="maxima")

[Out]

-7/400*sqrt(10)*log((3*x - sqrt(10) - 2)/(3*x + sqrt(10) - 2)) - 1/20*(7*x - 18)/(3*x^2 - 4*x - 2)

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mupad [B]  time = 0.21, size = 34, normalized size = 0.79 \[ \frac {7\,\sqrt {10}\,\mathrm {atanh}\left (\sqrt {10}\,\left (\frac {3\,x}{10}-\frac {1}{5}\right )\right )}{200}+\frac {\frac {7\,x}{60}-\frac {3}{10}}{-x^2+\frac {4\,x}{3}+\frac {2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x - 5)/(4*x - 3*x^2 + 2)^2,x)

[Out]

(7*10^(1/2)*atanh(10^(1/2)*((3*x)/10 - 1/5)))/200 + ((7*x)/60 - 3/10)/((4*x)/3 - x^2 + 2/3)

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sympy [A]  time = 0.17, size = 58, normalized size = 1.35 \[ - \frac {7 x - 18}{60 x^{2} - 80 x - 40} + \frac {7 \sqrt {10} \log {\left (x - \frac {2}{3} + \frac {\sqrt {10}}{3} \right )}}{400} - \frac {7 \sqrt {10} \log {\left (x - \frac {\sqrt {10}}{3} - \frac {2}{3} \right )}}{400} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-4*x)/(3*x**2-4*x-2)**2,x)

[Out]

-(7*x - 18)/(60*x**2 - 80*x - 40) + 7*sqrt(10)*log(x - 2/3 + sqrt(10)/3)/400 - 7*sqrt(10)*log(x - sqrt(10)/3 -
2/3)/400

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