3.105 \(\int \sqrt{-8+6 x+9 x^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{6} (3 x+1) \sqrt{9 x^2+6 x-8}-\frac{3}{2} \tanh ^{-1}\left (\frac{3 x+1}{\sqrt{9 x^2+6 x-8}}\right ) \]

[Out]

((1 + 3*x)*Sqrt[-8 + 6*x + 9*x^2])/6 - (3*ArcTanh[(1 + 3*x)/Sqrt[-8 + 6*x + 9*x^2]])/2

________________________________________________________________________________________

Rubi [A]  time = 0.0103741, antiderivative size = 49, 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, 621, 206} \[ \frac{1}{6} (3 x+1) \sqrt{9 x^2+6 x-8}-\frac{3}{2} \tanh ^{-1}\left (\frac{3 x+1}{\sqrt{9 x^2+6 x-8}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-8 + 6*x + 9*x^2],x]

[Out]

((1 + 3*x)*Sqrt[-8 + 6*x + 9*x^2])/6 - (3*ArcTanh[(1 + 3*x)/Sqrt[-8 + 6*x + 9*x^2]])/2

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 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 \sqrt{-8+6 x+9 x^2} \, dx &=\frac{1}{6} (1+3 x) \sqrt{-8+6 x+9 x^2}-\frac{9}{2} \int \frac{1}{\sqrt{-8+6 x+9 x^2}} \, dx\\ &=\frac{1}{6} (1+3 x) \sqrt{-8+6 x+9 x^2}-9 \operatorname{Subst}\left (\int \frac{1}{36-x^2} \, dx,x,\frac{6+18 x}{\sqrt{-8+6 x+9 x^2}}\right )\\ &=\frac{1}{6} (1+3 x) \sqrt{-8+6 x+9 x^2}-\frac{3}{2} \tanh ^{-1}\left (\frac{1+3 x}{\sqrt{-8+6 x+9 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0155534, size = 49, normalized size = 1. \[ \left (\frac{x}{2}+\frac{1}{6}\right ) \sqrt{9 x^2+6 x-8}-\frac{3}{2} \log \left (\sqrt{9 x^2+6 x-8}+3 x+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-8 + 6*x + 9*x^2],x]

[Out]

(1/6 + x/2)*Sqrt[-8 + 6*x + 9*x^2] - (3*Log[1 + 3*x + Sqrt[-8 + 6*x + 9*x^2]])/2

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 50, normalized size = 1. \begin{align*}{\frac{18\,x+6}{36}\sqrt{9\,{x}^{2}+6\,x-8}}-{\frac{\sqrt{9}}{2}\ln \left ({\frac{ \left ( 3+9\,x \right ) \sqrt{9}}{9}}+\sqrt{9\,{x}^{2}+6\,x-8} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^2+6*x-8)^(1/2),x)

[Out]

1/36*(18*x+6)*(9*x^2+6*x-8)^(1/2)-1/2*ln(1/9*(3+9*x)*9^(1/2)+(9*x^2+6*x-8)^(1/2))*9^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.4897, size = 70, normalized size = 1.43 \begin{align*} \frac{1}{2} \, \sqrt{9 \, x^{2} + 6 \, x - 8} x + \frac{1}{6} \, \sqrt{9 \, x^{2} + 6 \, x - 8} - \frac{3}{2} \, \log \left (18 \, x + 6 \, \sqrt{9 \, x^{2} + 6 \, x - 8} + 6\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2+6*x-8)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(9*x^2 + 6*x - 8)*x + 1/6*sqrt(9*x^2 + 6*x - 8) - 3/2*log(18*x + 6*sqrt(9*x^2 + 6*x - 8) + 6)

________________________________________________________________________________________

Fricas [A]  time = 2.16504, size = 109, normalized size = 2.22 \begin{align*} \frac{1}{6} \, \sqrt{9 \, x^{2} + 6 \, x - 8}{\left (3 \, x + 1\right )} + \frac{3}{2} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} + 6 \, x - 8} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2+6*x-8)^(1/2),x, algorithm="fricas")

[Out]

1/6*sqrt(9*x^2 + 6*x - 8)*(3*x + 1) + 3/2*log(-3*x + sqrt(9*x^2 + 6*x - 8) - 1)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{9 x^{2} + 6 x - 8}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**2+6*x-8)**(1/2),x)

[Out]

Integral(sqrt(9*x**2 + 6*x - 8), x)

________________________________________________________________________________________

Giac [A]  time = 1.21474, size = 55, normalized size = 1.12 \begin{align*} \frac{1}{6} \, \sqrt{9 \, x^{2} + 6 \, x - 8}{\left (3 \, x + 1\right )} + \frac{3}{2} \, \log \left ({\left | -3 \, x + \sqrt{9 \, x^{2} + 6 \, x - 8} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2+6*x-8)^(1/2),x, algorithm="giac")

[Out]

1/6*sqrt(9*x^2 + 6*x - 8)*(3*x + 1) + 3/2*log(abs(-3*x + sqrt(9*x^2 + 6*x - 8) - 1))