3.106 \(\int \sqrt{2+4 x+3 x^2} \, dx\)

Optimal. Leaf size=45 \[ \frac{1}{6} \sqrt{3 x^2+4 x+2} (3 x+2)+\frac{\sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{3 \sqrt{3}} \]

[Out]

((2 + 3*x)*Sqrt[2 + 4*x + 3*x^2])/6 + ArcSinh[(2 + 3*x)/Sqrt[2]]/(3*Sqrt[3])

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Rubi [A]  time = 0.0153222, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {612, 619, 215} \[ \frac{1}{6} \sqrt{3 x^2+4 x+2} (3 x+2)+\frac{\sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + 4*x + 3*x^2],x]

[Out]

((2 + 3*x)*Sqrt[2 + 4*x + 3*x^2])/6 + ArcSinh[(2 + 3*x)/Sqrt[2]]/(3*Sqrt[3])

Rule 612

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

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

Mathematica [A]  time = 0.0170433, size = 46, normalized size = 1.02 \[ \sqrt{3 x^2+4 x+2} \left (\frac{x}{2}+\frac{1}{3}\right )+\frac{\sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + 4*x + 3*x^2],x]

[Out]

(1/3 + x/2)*Sqrt[2 + 4*x + 3*x^2] + ArcSinh[(2 + 3*x)/Sqrt[2]]/(3*Sqrt[3])

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Maple [A]  time = 0.044, size = 35, normalized size = 0.8 \begin{align*}{\frac{4+6\,x}{12}\sqrt{3\,{x}^{2}+4\,x+2}}+{\frac{\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((3*x^2+4*x+2)^(1/2),x)

[Out]

1/12*(4+6*x)*(3*x^2+4*x+2)^(1/2)+1/9*3^(1/2)*arcsinh(3/2*2^(1/2)*(x+2/3))

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Maxima [A]  time = 1.70848, size = 62, normalized size = 1.38 \begin{align*} \frac{1}{2} \, \sqrt{3 \, x^{2} + 4 \, x + 2} x + \frac{1}{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((3*x^2+4*x+2)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(3*x^2 + 4*x + 2)*x + 1/9*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 = 1.96425, size = 158, normalized size = 3.51 \begin{align*} \frac{1}{6} \, \sqrt{3 \, x^{2} + 4 \, x + 2}{\left (3 \, x + 2\right )} + \frac{1}{18} \, \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3*x^2 + 4*x + 2)*(3*x + 2) + 1/18*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 4*x + 2)*(3*x + 2) - 9*x^2 - 12*x
- 5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(3*x**2 + 4*x + 2), x)

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Giac [A]  time = 1.2135, size = 72, normalized size = 1.6 \begin{align*} \frac{1}{6} \, \sqrt{3 \, x^{2} + 4 \, x + 2}{\left (3 \, x + 2\right )} - \frac{1}{9} \, \sqrt{3} \log \left (-\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x + 2}\right )} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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