3.123 \(\int \frac{1}{\sqrt{-2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{\tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x-2}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0204264, antiderivative size = 35, 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 \[ \frac{\tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x-2}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-2 + 5*x + 3*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.49536, size = 32, normalized size = 0.91 \[ \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x - 2}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**2+5*x-2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0155857, size = 28, normalized size = 0.8 \[ \frac{\log \left (2 \sqrt{9 x^2+15 x-6}+6 x+5\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[-2 + 5*x + 3*x^2],x]

[Out]

Log[5 + 6*x + 2*Sqrt[-6 + 15*x + 9*x^2]]/Sqrt[3]

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Maple [A]  time = 0.003, size = 30, normalized size = 0.9 \[{\frac{\sqrt{3}}{3}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x-2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^2+5*x-2)^(1/2),x)

[Out]

1/3*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 = 0.813478, size = 38, normalized size = 1.09 \[ \frac{1}{3} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x - 2} + 6 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^2 + 5*x - 2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.222686, size = 54, normalized size = 1.54 \[ \frac{1}{6} \, \sqrt{3} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 1\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x - 2}{\left (6 \, x + 5\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^2 + 5*x - 2),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log(sqrt(3)*(72*x^2 + 120*x + 1) + 12*sqrt(3*x^2 + 5*x - 2)*(6*x + 5
))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 x^{2} + 5 x - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x**2+5*x-2)**(1/2),x)

[Out]

Integral(1/sqrt(3*x**2 + 5*x - 2), x)

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GIAC/XCAS [A]  time = 0.215905, size = 46, normalized size = 1.31 \[ -\frac{1}{3} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x - 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^2 + 5*x - 2),x, algorithm="giac")

[Out]

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