### 3.2415 $$\int \frac{x}{\sqrt{-2+4 x+3 x^2}} \, dx$$

Optimal. Leaf size=54 $\frac{1}{3} \sqrt{3 x^2+4 x-2}-\frac{2 \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right )}{3 \sqrt{3}}$

[Out]

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

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Rubi [A]  time = 0.015042, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.188, Rules used = {640, 621, 206} $\frac{1}{3} \sqrt{3 x^2+4 x-2}-\frac{2 \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right )}{3 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x/Sqrt[-2 + 4*x + 3*x^2],x]

[Out]

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

Rule 640

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

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], 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{x}{\sqrt{-2+4 x+3 x^2}} \, dx &=\frac{1}{3} \sqrt{-2+4 x+3 x^2}-\frac{2}{3} \int \frac{1}{\sqrt{-2+4 x+3 x^2}} \, dx\\ &=\frac{1}{3} \sqrt{-2+4 x+3 x^2}-\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{4+6 x}{\sqrt{-2+4 x+3 x^2}}\right )\\ &=\frac{1}{3} \sqrt{-2+4 x+3 x^2}-\frac{2 \tanh ^{-1}\left (\frac{2+3 x}{\sqrt{3} \sqrt{-2+4 x+3 x^2}}\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0167977, size = 49, normalized size = 0.91 $\frac{1}{9} \left (3 \sqrt{3 x^2+4 x-2}-2 \sqrt{3} \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{9 x^2+12 x-6}}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/Sqrt[-2 + 4*x + 3*x^2],x]

[Out]

(3*Sqrt[-2 + 4*x + 3*x^2] - 2*Sqrt[3]*ArcTanh[(2 + 3*x)/Sqrt[-6 + 12*x + 9*x^2]])/9

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Maple [A]  time = 0.041, size = 45, normalized size = 0.8 \begin{align*}{\frac{1}{3}\sqrt{3\,{x}^{2}+4\,x-2}}-{\frac{2\,\sqrt{3}}{9}\ln \left ({\frac{ \left ( 2+3\,x \right ) \sqrt{3}}{3}}+\sqrt{3\,{x}^{2}+4\,x-2} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x/(3*x^2+4*x-2)^(1/2),x)

[Out]

1/3*(3*x^2+4*x-2)^(1/2)-2/9*ln(1/3*(2+3*x)*3^(1/2)+(3*x^2+4*x-2)^(1/2))*3^(1/2)

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Maxima [A]  time = 1.48325, size = 58, normalized size = 1.07 \begin{align*} -\frac{2}{9} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 4 \, x - 2} + 6 \, x + 4\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x - 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^2+4*x-2)^(1/2),x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 4*x - 2) + 6*x + 4) + 1/3*sqrt(3*x^2 + 4*x - 2)

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Fricas [A]  time = 2.45922, size = 143, normalized size = 2.65 \begin{align*} \frac{1}{9} \, \sqrt{3} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 4 \, x - 2}{\left (3 \, x + 2\right )} + 9 \, x^{2} + 12 \, x - 1\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x - 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^2+4*x-2)^(1/2),x, algorithm="fricas")

[Out]

1/9*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 4*x - 2)*(3*x + 2) + 9*x^2 + 12*x - 1) + 1/3*sqrt(3*x^2 + 4*x - 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{3 x^{2} + 4 x - 2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x**2+4*x-2)**(1/2),x)

[Out]

Integral(x/sqrt(3*x**2 + 4*x - 2), x)

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Giac [A]  time = 1.17741, size = 66, normalized size = 1.22 \begin{align*} \frac{2}{9} \, \sqrt{3} \log \left ({\left | -\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x - 2}\right )} - 2 \right |}\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x - 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^2+4*x-2)^(1/2),x, algorithm="giac")

[Out]

2/9*sqrt(3)*log(abs(-sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 4*x - 2)) - 2)) + 1/3*sqrt(3*x^2 + 4*x - 2)