3.200 \(\int \frac{-3+2 x}{(3+6 x+2 x^2)^3} \, dx\)

Optimal. Leaf size=61 \[ -\frac{2 x+3}{2 \left (2 x^2+6 x+3\right )}+\frac{4 x+5}{4 \left (2 x^2+6 x+3\right )^2}+\frac{\tanh ^{-1}\left (\frac{2 x+3}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

(5 + 4*x)/(4*(3 + 6*x + 2*x^2)^2) - (3 + 2*x)/(2*(3 + 6*x + 2*x^2)) + ArcTanh[(3 + 2*x)/Sqrt[3]]/Sqrt[3]

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Rubi [A]  time = 0.0221754, antiderivative size = 61, 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 = {638, 614, 618, 206} \[ -\frac{2 x+3}{2 \left (2 x^2+6 x+3\right )}+\frac{4 x+5}{4 \left (2 x^2+6 x+3\right )^2}+\frac{\tanh ^{-1}\left (\frac{2 x+3}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(-3 + 2*x)/(3 + 6*x + 2*x^2)^3,x]

[Out]

(5 + 4*x)/(4*(3 + 6*x + 2*x^2)^2) - (3 + 2*x)/(2*(3 + 6*x + 2*x^2)) + ArcTanh[(3 + 2*x)/Sqrt[3]]/Sqrt[3]

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 614

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

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 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])

Rubi steps

\begin{align*} \int \frac{-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx &=\frac{5+4 x}{4 \left (3+6 x+2 x^2\right )^2}+3 \int \frac{1}{\left (3+6 x+2 x^2\right )^2} \, dx\\ &=\frac{5+4 x}{4 \left (3+6 x+2 x^2\right )^2}-\frac{3+2 x}{2 \left (3+6 x+2 x^2\right )}-\int \frac{1}{3+6 x+2 x^2} \, dx\\ &=\frac{5+4 x}{4 \left (3+6 x+2 x^2\right )^2}-\frac{3+2 x}{2 \left (3+6 x+2 x^2\right )}+2 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,6+4 x\right )\\ &=\frac{5+4 x}{4 \left (3+6 x+2 x^2\right )^2}-\frac{3+2 x}{2 \left (3+6 x+2 x^2\right )}+\frac{\tanh ^{-1}\left (\frac{3+2 x}{\sqrt{3}}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.04588, size = 70, normalized size = 1.15 \[ \frac{1}{12} \left (-\frac{3 \left (8 x^3+36 x^2+44 x+13\right )}{\left (2 x^2+6 x+3\right )^2}-2 \sqrt{3} \log \left (-2 x+\sqrt{3}-3\right )+2 \sqrt{3} \log \left (2 x+\sqrt{3}+3\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 2*x)/(3 + 6*x + 2*x^2)^3,x]

[Out]

((-3*(13 + 44*x + 36*x^2 + 8*x^3))/(3 + 6*x + 2*x^2)^2 - 2*Sqrt[3]*Log[-3 + Sqrt[3] - 2*x] + 2*Sqrt[3]*Log[3 +
Sqrt[3] + 2*x])/12

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Maple [A]  time = 0.002, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-24\,x-30}{24\, \left ( 2\,{x}^{2}+6\,x+3 \right ) ^{2}}}-{\frac{6+4\,x}{8\,{x}^{2}+24\,x+12}}+{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\frac{ \left ( 6+4\,x \right ) \sqrt{3}}{6}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3+2*x)/(2*x^2+6*x+3)^3,x)

[Out]

-1/24*(-24*x-30)/(2*x^2+6*x+3)^2-1/4*(6+4*x)/(2*x^2+6*x+3)+1/3*3^(1/2)*arctanh(1/6*(6+4*x)*3^(1/2))

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Maxima [A]  time = 1.41719, size = 90, normalized size = 1.48 \begin{align*} -\frac{1}{6} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3} + 3}{2 \, x + \sqrt{3} + 3}\right ) - \frac{8 \, x^{3} + 36 \, x^{2} + 44 \, x + 13}{4 \,{\left (4 \, x^{4} + 24 \, x^{3} + 48 \, x^{2} + 36 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+2*x)/(2*x^2+6*x+3)^3,x, algorithm="maxima")

[Out]

-1/6*sqrt(3)*log((2*x - sqrt(3) + 3)/(2*x + sqrt(3) + 3)) - 1/4*(8*x^3 + 36*x^2 + 44*x + 13)/(4*x^4 + 24*x^3 +
48*x^2 + 36*x + 9)

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Fricas [A]  time = 1.62373, size = 251, normalized size = 4.11 \begin{align*} -\frac{24 \, x^{3} - 2 \, \sqrt{3}{\left (4 \, x^{4} + 24 \, x^{3} + 48 \, x^{2} + 36 \, x + 9\right )} \log \left (\frac{2 \, x^{2} + \sqrt{3}{\left (2 \, x + 3\right )} + 6 \, x + 6}{2 \, x^{2} + 6 \, x + 3}\right ) + 108 \, x^{2} + 132 \, x + 39}{12 \,{\left (4 \, x^{4} + 24 \, x^{3} + 48 \, x^{2} + 36 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+2*x)/(2*x^2+6*x+3)^3,x, algorithm="fricas")

[Out]

-1/12*(24*x^3 - 2*sqrt(3)*(4*x^4 + 24*x^3 + 48*x^2 + 36*x + 9)*log((2*x^2 + sqrt(3)*(2*x + 3) + 6*x + 6)/(2*x^
2 + 6*x + 3)) + 108*x^2 + 132*x + 39)/(4*x^4 + 24*x^3 + 48*x^2 + 36*x + 9)

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Sympy [A]  time = 0.146863, size = 75, normalized size = 1.23 \begin{align*} - \frac{8 x^{3} + 36 x^{2} + 44 x + 13}{16 x^{4} + 96 x^{3} + 192 x^{2} + 144 x + 36} - \frac{\sqrt{3} \log{\left (x - \frac{\sqrt{3}}{2} + \frac{3}{2} \right )}}{6} + \frac{\sqrt{3} \log{\left (x + \frac{\sqrt{3}}{2} + \frac{3}{2} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+2*x)/(2*x**2+6*x+3)**3,x)

[Out]

-(8*x**3 + 36*x**2 + 44*x + 13)/(16*x**4 + 96*x**3 + 192*x**2 + 144*x + 36) - sqrt(3)*log(x - sqrt(3)/2 + 3/2)
/6 + sqrt(3)*log(x + sqrt(3)/2 + 3/2)/6

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Giac [A]  time = 1.06024, size = 82, normalized size = 1.34 \begin{align*} -\frac{1}{6} \, \sqrt{3} \log \left (\frac{{\left | 4 \, x - 2 \, \sqrt{3} + 6 \right |}}{{\left | 4 \, x + 2 \, \sqrt{3} + 6 \right |}}\right ) - \frac{8 \, x^{3} + 36 \, x^{2} + 44 \, x + 13}{4 \,{\left (2 \, x^{2} + 6 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+2*x)/(2*x^2+6*x+3)^3,x, algorithm="giac")

[Out]

-1/6*sqrt(3)*log(abs(4*x - 2*sqrt(3) + 6)/abs(4*x + 2*sqrt(3) + 6)) - 1/4*(8*x^3 + 36*x^2 + 44*x + 13)/(2*x^2
+ 6*x + 3)^2