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

Optimal. Leaf size=34 \[ -\frac{2 \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]

[Out]

(-2*ArcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c]

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Rubi [A]  time = 0.0378399, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Int[(c + b*x + a*x^2)^(-1),x]

[Out]

(-2*ArcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c]

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Rubi in Sympy [A]  time = 2.05828, size = 34, normalized size = 1. \[ - \frac{2 \operatorname{atanh}{\left (\frac{2 a x + b}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atanh((2*a*x + b)/sqrt(-4*a*c + b**2))/sqrt(-4*a*c + b**2)

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Mathematica [A]  time = 0.052531, size = 38, normalized size = 1.12 \[ \frac{2 \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + b*x + a*x^2)^(-1),x]

[Out]

(2*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]

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Maple [A]  time = 0.005, size = 35, normalized size = 1. \[ 2\,{\frac{1}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x^2+b*x+c),x)

[Out]

2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(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/(a*x^2 + b*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.198196, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} x -{\left (2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{a x^{2} + b x + c}\right )}{\sqrt{b^{2} - 4 \, a c}}, \frac{2 \, \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right )}{\sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[log(-(b^3 - 4*a*b*c + 2*(a*b^2 - 4*a^2*c)*x - (2*a^2*x^2 + 2*a*b*x + b^2 - 2*a*
c)*sqrt(b^2 - 4*a*c))/(a*x^2 + b*x + c))/sqrt(b^2 - 4*a*c), 2*arctan(-sqrt(-b^2
+ 4*a*c)*(2*a*x + b)/(b^2 - 4*a*c))/sqrt(-b^2 + 4*a*c)]

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Sympy [A]  time = 0.258405, size = 124, normalized size = 3.65 \[ - \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{- 4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b}{2 a} \right )} + \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b}{2 a} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x**2+b*x+c),x)

[Out]

-sqrt(-1/(4*a*c - b**2))*log(x + (-4*a*c*sqrt(-1/(4*a*c - b**2)) + b**2*sqrt(-1/
(4*a*c - b**2)) + b)/(2*a)) + sqrt(-1/(4*a*c - b**2))*log(x + (4*a*c*sqrt(-1/(4*
a*c - b**2)) - b**2*sqrt(-1/(4*a*c - b**2)) + b)/(2*a))

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GIAC/XCAS [A]  time = 0.212984, size = 46, normalized size = 1.35 \[ \frac{2 \, \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan((2*a*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c)