3.103 \(\int \sqrt{5-6 x+9 x^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{2}{3} \sinh ^{-1}\left (\frac{1}{2} (3 x-1)\right )-\frac{1}{6} (1-3 x) \sqrt{9 x^2-6 x+5} \]

[Out]

-((1 - 3*x)*Sqrt[5 - 6*x + 9*x^2])/6 + (2*ArcSinh[(-1 + 3*x)/2])/3

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Rubi [A]  time = 0.0113981, antiderivative size = 38, 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} \[ \frac{2}{3} \sinh ^{-1}\left (\frac{1}{2} (3 x-1)\right )-\frac{1}{6} (1-3 x) \sqrt{9 x^2-6 x+5} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[5 - 6*x + 9*x^2],x]

[Out]

-((1 - 3*x)*Sqrt[5 - 6*x + 9*x^2])/6 + (2*ArcSinh[(-1 + 3*x)/2])/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{5-6 x+9 x^2} \, dx &=-\frac{1}{6} (1-3 x) \sqrt{5-6 x+9 x^2}+2 \int \frac{1}{\sqrt{5-6 x+9 x^2}} \, dx\\ &=-\frac{1}{6} (1-3 x) \sqrt{5-6 x+9 x^2}+\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{144}}} \, dx,x,-6+18 x\right )\\ &=-\frac{1}{6} (1-3 x) \sqrt{5-6 x+9 x^2}+\frac{2}{3} \sinh ^{-1}\left (\frac{1}{2} (-1+3 x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0165592, size = 39, normalized size = 1.03 \[ \sqrt{9 x^2-6 x+5} \left (\frac{x}{2}-\frac{1}{6}\right )+\frac{2}{3} \sinh ^{-1}\left (\frac{1}{2} (3 x-1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[5 - 6*x + 9*x^2],x]

[Out]

(-1/6 + x/2)*Sqrt[5 - 6*x + 9*x^2] + (2*ArcSinh[(-1 + 3*x)/2])/3

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Maple [A]  time = 0.049, size = 29, normalized size = 0.8 \begin{align*}{\frac{18\,x-6}{36}\sqrt{9\,{x}^{2}-6\,x+5}}+{\frac{2}{3}{\it Arcsinh} \left ( -{\frac{1}{2}}+{\frac{3\,x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^2-6*x+5)^(1/2),x)

[Out]

1/36*(18*x-6)*(9*x^2-6*x+5)^(1/2)+2/3*arcsinh(-1/2+3/2*x)

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Maxima [A]  time = 1.61918, size = 51, normalized size = 1.34 \begin{align*} \frac{1}{2} \, \sqrt{9 \, x^{2} - 6 \, x + 5} x - \frac{1}{6} \, \sqrt{9 \, x^{2} - 6 \, x + 5} + \frac{2}{3} \, \operatorname{arsinh}\left (\frac{3}{2} \, x - \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-6*x+5)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(9*x^2 - 6*x + 5)*x - 1/6*sqrt(9*x^2 - 6*x + 5) + 2/3*arcsinh(3/2*x - 1/2)

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Fricas [A]  time = 2.40969, size = 109, normalized size = 2.87 \begin{align*} \frac{1}{6} \, \sqrt{9 \, x^{2} - 6 \, x + 5}{\left (3 \, x - 1\right )} - \frac{2}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} - 6 \, x + 5} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-6*x+5)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{9 x^{2} - 6 x + 5}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**2-6*x+5)**(1/2),x)

[Out]

Integral(sqrt(9*x**2 - 6*x + 5), x)

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Giac [A]  time = 1.20949, size = 54, normalized size = 1.42 \begin{align*} \frac{1}{6} \, \sqrt{9 \, x^{2} - 6 \, x + 5}{\left (3 \, x - 1\right )} - \frac{2}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} - 6 \, x + 5} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-6*x+5)^(1/2),x, algorithm="giac")

[Out]

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