3.2.6 \(\int \sqrt {2+4 x+3 x^2} \, dx\) [106]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 = {626, 633, 221} \begin {gather*} \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}} \end {gather*}

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 221

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]

Rule 626

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 633

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]

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 {\text {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*}

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Mathematica [A]
time = 0.08, size = 53, normalized size = 1.18 \begin {gather*} \frac {1}{6} (2+3 x) \sqrt {2+4 x+3 x^2}-\frac {\log \left (-2-3 x+\sqrt {6+12 x+9 x^2}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + 4*x + 3*x^2],x]

[Out]

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

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Maple [A]
time = 0.61, size = 35, normalized size = 0.78

method	result	size
default	\(\frac {\left (6 x +4\right ) \sqrt {3 x^{2}+4 x +2}}{12}+\frac {\sqrt {3}\, \arcsinh \left (\frac {3 \sqrt {2}\, \left (x +\frac {2}{3}\right )}{2}\right )}{9}\)	\(35\)
risch	\(\frac {\left (2+3 x \right ) \sqrt {3 x^{2}+4 x +2}}{6}+\frac {\sqrt {3}\, \arcsinh \left (\frac {3 \sqrt {2}\, \left (x +\frac {2}{3}\right )}{2}\right )}{9}\)	\(35\)
trager	\(\left (\frac {1}{3}+\frac {x}{2}\right ) \sqrt {3 x^{2}+4 x +2}+\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +3 \sqrt {3 x^{2}+4 x +2}+2 \RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{9}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+4*x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/12*(6*x+4)*(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 = 0.51, size = 46, normalized size = 1.02 \begin {gather*} \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 {gather*}

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 = 1.55, size = 58, normalized size = 1.29 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {3 x^{2} + 4 x + 2}\, dx \end {gather*}

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 = 1.01, size = 53, normalized size = 1.18 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.19, size = 48, normalized size = 1.07 \begin {gather*} \frac {\sqrt {3}\,\ln \left (\sqrt {3\,x^2+4\,x+2}+\frac {\sqrt {3}\,\left (3\,x+2\right )}{3}\right )}{9}+\left (\frac {x}{2}+\frac {1}{3}\right )\,\sqrt {3\,x^2+4\,x+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + 3*x^2 + 2)^(1/2),x)

[Out]

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

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