3.168 \(\int \frac{-4+7 x}{(5+2 x+3 x^2)^2} \, dx\)

Optimal. Leaf size=43 \[ -\frac{19 x+39}{28 \left (3 x^2+2 x+5\right )}-\frac{19 \tan ^{-1}\left (\frac{3 x+1}{\sqrt{14}}\right )}{28 \sqrt{14}} \]

[Out]

-(39 + 19*x)/(28*(5 + 2*x + 3*x^2)) - (19*ArcTan[(1 + 3*x)/Sqrt[14]])/(28*Sqrt[14])

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Rubi [A]  time = 0.0206964, antiderivative size = 43, normalized size of antiderivative = 1., 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, 204} \[ -\frac{19 x+39}{28 \left (3 x^2+2 x+5\right )}-\frac{19 \tan ^{-1}\left (\frac{3 x+1}{\sqrt{14}}\right )}{28 \sqrt{14}} \]

Antiderivative was successfully verified.

[In]

Int[(-4 + 7*x)/(5 + 2*x + 3*x^2)^2,x]

[Out]

-(39 + 19*x)/(28*(5 + 2*x + 3*x^2)) - (19*ArcTan[(1 + 3*x)/Sqrt[14]])/(28*Sqrt[14])

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]

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])

Rubi steps

\begin{align*} \int \frac{-4+7 x}{\left (5+2 x+3 x^2\right )^2} \, dx &=-\frac{39+19 x}{28 \left (5+2 x+3 x^2\right )}-\frac{19}{28} \int \frac{1}{5+2 x+3 x^2} \, dx\\ &=-\frac{39+19 x}{28 \left (5+2 x+3 x^2\right )}+\frac{19}{14} \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+6 x\right )\\ &=-\frac{39+19 x}{28 \left (5+2 x+3 x^2\right )}-\frac{19 \tan ^{-1}\left (\frac{1+3 x}{\sqrt{14}}\right )}{28 \sqrt{14}}\\ \end{align*}

Mathematica [A]  time = 0.033955, size = 43, normalized size = 1. \[ \frac{-19 x-39}{28 \left (3 x^2+2 x+5\right )}-\frac{19 \tan ^{-1}\left (\frac{3 x+1}{\sqrt{14}}\right )}{28 \sqrt{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 7*x)/(5 + 2*x + 3*x^2)^2,x]

[Out]

(-39 - 19*x)/(28*(5 + 2*x + 3*x^2)) - (19*ArcTan[(1 + 3*x)/Sqrt[14]])/(28*Sqrt[14])

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Maple [A]  time = 0.004, size = 37, normalized size = 0.9 \begin{align*}{\frac{-38\,x-78}{168\,{x}^{2}+112\,x+280}}-{\frac{19\,\sqrt{14}}{392}\arctan \left ({\frac{ \left ( 6\,x+2 \right ) \sqrt{14}}{28}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4+7*x)/(3*x^2+2*x+5)^2,x)

[Out]

1/56*(-38*x-78)/(3*x^2+2*x+5)-19/392*14^(1/2)*arctan(1/28*(6*x+2)*14^(1/2))

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Maxima [A]  time = 1.40944, size = 49, normalized size = 1.14 \begin{align*} -\frac{19}{392} \, \sqrt{14} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (3 \, x + 1\right )}\right ) - \frac{19 \, x + 39}{28 \,{\left (3 \, x^{2} + 2 \, x + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4+7*x)/(3*x^2+2*x+5)^2,x, algorithm="maxima")

[Out]

-19/392*sqrt(14)*arctan(1/14*sqrt(14)*(3*x + 1)) - 1/28*(19*x + 39)/(3*x^2 + 2*x + 5)

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Fricas [A]  time = 1.96407, size = 140, normalized size = 3.26 \begin{align*} -\frac{19 \, \sqrt{14}{\left (3 \, x^{2} + 2 \, x + 5\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (3 \, x + 1\right )}\right ) + 266 \, x + 546}{392 \,{\left (3 \, x^{2} + 2 \, x + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4+7*x)/(3*x^2+2*x+5)^2,x, algorithm="fricas")

[Out]

-1/392*(19*sqrt(14)*(3*x^2 + 2*x + 5)*arctan(1/14*sqrt(14)*(3*x + 1)) + 266*x + 546)/(3*x^2 + 2*x + 5)

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Sympy [A]  time = 0.128665, size = 42, normalized size = 0.98 \begin{align*} - \frac{19 x + 39}{84 x^{2} + 56 x + 140} - \frac{19 \sqrt{14} \operatorname{atan}{\left (\frac{3 \sqrt{14} x}{14} + \frac{\sqrt{14}}{14} \right )}}{392} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4+7*x)/(3*x**2+2*x+5)**2,x)

[Out]

-(19*x + 39)/(84*x**2 + 56*x + 140) - 19*sqrt(14)*atan(3*sqrt(14)*x/14 + sqrt(14)/14)/392

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Giac [A]  time = 1.06648, size = 49, normalized size = 1.14 \begin{align*} -\frac{19}{392} \, \sqrt{14} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (3 \, x + 1\right )}\right ) - \frac{19 \, x + 39}{28 \,{\left (3 \, x^{2} + 2 \, x + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4+7*x)/(3*x^2+2*x+5)^2,x, algorithm="giac")

[Out]

-19/392*sqrt(14)*arctan(1/14*sqrt(14)*(3*x + 1)) - 1/28*(19*x + 39)/(3*x^2 + 2*x + 5)