3.288 \(\int \frac{1}{\sqrt{-5+12 x+9 x^2}} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{3} \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{9 x^2+12 x-5}}\right ) \]

[Out]

ArcTanh[(2 + 3*x)/Sqrt[-5 + 12*x + 9*x^2]]/3

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Rubi [A]  time = 0.0139653, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{3} \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{9 x^2+12 x-5}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-5 + 12*x + 9*x^2],x]

[Out]

ArcTanh[(2 + 3*x)/Sqrt[-5 + 12*x + 9*x^2]]/3

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Rubi in Sympy [A]  time = 0.697248, size = 22, normalized size = 0.88 \[ \frac{\operatorname{atanh}{\left (\frac{18 x + 12}{6 \sqrt{9 x^{2} + 12 x - 5}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(9*x**2+12*x-5)**(1/2),x)

[Out]

atanh((18*x + 12)/(6*sqrt(9*x**2 + 12*x - 5)))/3

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Mathematica [A]  time = 0.00937678, size = 24, normalized size = 0.96 \[ \frac{1}{3} \log \left (\sqrt{9 x^2+12 x-5}+3 x+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[-5 + 12*x + 9*x^2],x]

[Out]

Log[2 + 3*x + Sqrt[-5 + 12*x + 9*x^2]]/3

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Maple [A]  time = 0.003, size = 30, normalized size = 1.2 \[{\frac{\sqrt{9}}{9}\ln \left ({\frac{ \left ( 6+9\,x \right ) \sqrt{9}}{9}}+\sqrt{9\,{x}^{2}+12\,x-5} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(9*x^2+12*x-5)^(1/2),x)

[Out]

1/9*ln(1/9*(6+9*x)*9^(1/2)+(9*x^2+12*x-5)^(1/2))*9^(1/2)

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Maxima [A]  time = 1.50057, size = 30, normalized size = 1.2 \[ \frac{1}{3} \, \log \left (18 \, x + 6 \, \sqrt{9 \, x^{2} + 12 \, x - 5} + 12\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(9*x^2 + 12*x - 5),x, algorithm="maxima")

[Out]

1/3*log(18*x + 6*sqrt(9*x^2 + 12*x - 5) + 12)

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Fricas [A]  time = 0.20906, size = 27, normalized size = 1.08 \[ -\frac{1}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} + 12 \, x - 5} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(9*x^2 + 12*x - 5),x, algorithm="fricas")

[Out]

-1/3*log(-3*x + sqrt(9*x^2 + 12*x - 5) - 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{9 x^{2} + 12 x - 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(9*x**2+12*x-5)**(1/2),x)

[Out]

Integral(1/sqrt(9*x**2 + 12*x - 5), x)

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GIAC/XCAS [A]  time = 0.221044, size = 28, normalized size = 1.12 \[ -\frac{1}{3} \,{\rm ln}\left ({\left | -3 \, x + \sqrt{9 \, x^{2} + 12 \, x - 5} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(9*x^2 + 12*x - 5),x, algorithm="giac")

[Out]

-1/3*ln(abs(-3*x + sqrt(9*x^2 + 12*x - 5) - 2))