Optimal. Leaf size=34 \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]
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Rubi [A] time = 0.0438729, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 4.39266, size = 34, normalized size = 1. \[ - \frac{2 \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.0113392, size = 38, normalized size = 1.12 \[ \frac{2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(-1),x]
[Out]
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Maple [A] time = 0.003, size = 35, normalized size = 1. \[ 2\,{\frac{1}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x^2+b*x+a),x)
[Out]
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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/(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.261094, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right )}{\sqrt{b^{2} - 4 \, a c}}, \frac{2 \, \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{\sqrt{-b^{2} + 4 \, a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.565083, size = 124, normalized size = 3.65 \[ - \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{- 4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b}{2 c} \right )} + \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b}{2 c} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.204647, size = 46, normalized size = 1.35 \[ \frac{2 \, \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x^2 + b*x + a),x, algorithm="giac")
[Out]