3.2366 \(\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=36 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c}} \]

[Out]

ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/Sqrt[c]

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Rubi [A]  time = 0.0294308, antiderivative size = 36, 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 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a + b*x + c*x^2],x]

[Out]

ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/Sqrt[c]

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Rubi in Sympy [A]  time = 2.8439, size = 32, normalized size = 0.89 \[ \frac{\operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x+a)**(1/2),x)

[Out]

atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/sqrt(c)

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Mathematica [A]  time = 0.0340967, size = 34, normalized size = 0.94 \[ \frac{\log \left (2 \sqrt{c} \sqrt{a+b x+c x^2}+b+2 c x\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a + b*x + c*x^2],x]

[Out]

Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]]/Sqrt[c]

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Maple [A]  time = 0.004, size = 30, normalized size = 0.8 \[{1\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x+a)^(1/2),x)

[Out]

ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238499, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{2 \, \sqrt{c}}, \frac{\arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{\sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

[1/2*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 +
4*a*c)*sqrt(c))/sqrt(c), arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)
*c))/sqrt(-c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(1/sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 0.21804, size = 49, normalized size = 1.36 \[ -\frac{{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

-ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/sqrt(c)