### 3.2411 $$\int \frac{x}{\sqrt{2+4 x+3 x^2}} \, dx$$

Optimal. Leaf size=40 $\frac{1}{3} \sqrt{3 x^2+4 x+2}-\frac{2 \sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{3 \sqrt{3}}$

[Out]

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

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Rubi [A]  time = 0.0177499, antiderivative size = 40, 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, 619, 215} $\frac{1}{3} \sqrt{3 x^2+4 x+2}-\frac{2 \sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{3 \sqrt{3}}$

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*ArcSinh[(2 + 3*x)/Sqrt[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 619

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

Rule 215

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

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{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{8}}} \, dx,x,4+6 x\right )}{3 \sqrt{6}}\\ &=\frac{1}{3} \sqrt{2+4 x+3 x^2}-\frac{2 \sinh ^{-1}\left (\frac{2+3 x}{\sqrt{2}}\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0123403, size = 40, normalized size = 1. $\frac{1}{9} \left (3 \sqrt{3 x^2+4 x+2}-2 \sqrt{3} \sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )\right )$

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]*ArcSinh[(2 + 3*x)/Sqrt[2]])/9

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Maple [A]  time = 0.043, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{3}\sqrt{3\,{x}^{2}+4\,x+2}}-{\frac{2\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{3\,\sqrt{2}}{2} \left ( x+{\frac{2}{3}} \right ) } \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*3^(1/2)*arcsinh(3/2*2^(1/2)*(x+2/3))

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Maxima [A]  time = 1.47642, size = 42, normalized size = 1.05 \begin{align*} -\frac{2}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 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="maxima")

[Out]

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

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Fricas [A]  time = 2.02219, size = 142, normalized size = 3.55 \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 - 5\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 - 5) + 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 = 1.17975, size = 65, normalized size = 1.62 \begin{align*} \frac{2}{9} \, \sqrt{3} \log \left (-\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x + 2}\right )} - 2\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(-sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 4*x + 2)) - 2) + 1/3*sqrt(3*x^2 + 4*x + 2)