3.2.5 \(\int \sqrt {-8+6 x+9 x^2} \, dx\) [105]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 49, 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, 635, 212} \begin {gather*} \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 ) \end {gather*}

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 635

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]

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 \text {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*}

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Mathematica [A]
time = 0.07, size = 49, normalized size = 1.00 \begin {gather*} \frac {1}{6} (1+3 x) \sqrt {-8+6 x+9 x^2}-3 \tanh ^{-1}\left (\frac {\sqrt {-8+6 x+9 x^2}}{-2+3 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-8 + 6*x + 9*x^2],x]

[Out]

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

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Maple [A]
time = 0.49, size = 50, normalized size = 1.02

method	result	size
trager	\(\left (\frac {x}{2}+\frac {1}{6}\right ) \sqrt {9 x^{2}+6 x -8}-\frac {3 \ln \left (\sqrt {9 x^{2}+6 x -8}+1+3 x \right )}{2}\)	\(40\)
default	\(\frac {\left (18 x +6\right ) \sqrt {9 x^{2}+6 x -8}}{36}-\frac {\ln \left (\frac {\left (9 x +3\right ) \sqrt {9}}{9}+\sqrt {9 x^{2}+6 x -8}\right ) \sqrt {9}}{2}\)	\(50\)
risch	\(\frac {\left (3 x +1\right ) \sqrt {9 x^{2}+6 x -8}}{6}-\frac {\ln \left (\frac {\left (9 x +3\right ) \sqrt {9}}{9}+\sqrt {9 x^{2}+6 x -8}\right ) \sqrt {9}}{2}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.54, size = 52, normalized size = 1.06 \begin {gather*} \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 {gather*}

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)

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Fricas [A]
time = 1.26, size = 40, normalized size = 0.82 \begin {gather*} \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 {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {9 x^{2} + 6 x - 8}\, dx \end {gather*}

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)

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Giac [A]
time = 1.46, size = 41, normalized size = 0.84 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 0.21, size = 39, normalized size = 0.80 \begin {gather*} \left (\frac {x}{2}+\frac {1}{6}\right )\,\sqrt {9\,x^2+6\,x-8}-\frac {3\,\ln \left (3\,x+\sqrt {9\,x^2+6\,x-8}+1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + 9*x^2 - 8)^(1/2),x)

[Out]

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

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