3.2413 \(\int \frac{x}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=57 \[ \frac{1}{3} \sqrt{3 x^2+5 x+2}-\frac{5 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{6 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.014573, antiderivative size = 57, 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} \[ \frac{1}{3} \sqrt{3 x^2+5 x+2}-\frac{5 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[2 + 5*x + 3*x^2],x]

[Out]

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

Mathematica [A]  time = 0.0186334, size = 52, normalized size = 0.91 \[ \frac{1}{18} \left (6 \sqrt{3 x^2+5 x+2}-5 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(6*Sqrt[2 + 5*x + 3*x^2] - 5*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/18

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Maple [A]  time = 0.041, size = 45, normalized size = 0.8 \begin{align*}{\frac{1}{3}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{5\,\sqrt{3}}{18}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(3*x^2+5*x+2)^(1/2),x)

[Out]

1/3*(3*x^2+5*x+2)^(1/2)-5/18*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)

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Maxima [A]  time = 1.57609, size = 58, normalized size = 1.02 \begin{align*} -\frac{5}{18} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.08253, size = 151, normalized size = 2.65 \begin{align*} \frac{5}{36} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

5/36*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 1/3*sqrt(3*x^2 + 5*x + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{\left (x + 1\right ) \left (3 x + 2\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x**2+5*x+2)**(1/2),x)

[Out]

Integral(x/sqrt((x + 1)*(3*x + 2)), x)

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Giac [A]  time = 1.16071, size = 66, normalized size = 1.16 \begin{align*} \frac{5}{18} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

5/18*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 1/3*sqrt(3*x^2 + 5*x + 2)