Optimal. Leaf size=70 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b x+c x^2\right )}{2 c^2}+\frac{x}{c} \]
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Rubi [A] time = 0.105374, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b x+c x^2\right )}{2 c^2}+\frac{x}{c} \]
Antiderivative was successfully verified.
[In] Int[(c + a/x^2 + b/x)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 21.1567, size = 65, normalized size = 0.93 \[ - \frac{b \log{\left (a + b x + c x^{2} \right )}}{2 c^{2}} + \frac{x}{c} - \frac{\left (- 2 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c+a/x**2+b/x),x)
[Out]
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Mathematica [A] time = 0.103111, size = 73, normalized size = 1.04 \[ \frac{\left (b^2-2 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{c^2 \sqrt{4 a c-b^2}}-\frac{b \log \left (a+b x+c x^2\right )}{2 c^2}+\frac{x}{c} \]
Antiderivative was successfully verified.
[In] Integrate[(c + a/x^2 + b/x)^(-1),x]
[Out]
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Maple [A] time = 0.004, size = 101, normalized size = 1.4 \[{\frac{x}{c}}-{\frac{b\ln \left ( c{x}^{2}+bx+a \right ) }{2\,{c}^{2}}}-2\,{\frac{a}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}}{{c}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c+a/x^2+b/x),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 + b/x + a/x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280536, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 2 \, a c\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x - b \log \left (c x^{2} + b x + a\right )\right )}}{2 \, \sqrt{b^{2} - 4 \, a c} c^{2}}, \frac{2 \,{\left (b^{2} - 2 \, a c\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x - b \log \left (c x^{2} + b x + a\right )\right )}}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c + b/x + a/x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.37063, size = 306, normalized size = 4.37 \[ \left (- \frac{b}{2 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- a b - 4 a c^{2} \left (- \frac{b}{2 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + b^{2} c \left (- \frac{b}{2 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (- \frac{b}{2 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- a b - 4 a c^{2} \left (- \frac{b}{2 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + b^{2} c \left (- \frac{b}{2 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac{x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c+a/x**2+b/x),x)
[Out]
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GIAC/XCAS [A] time = 0.290691, size = 90, normalized size = 1.29 \[ \frac{x}{c} - \frac{b{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac{{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c + b/x + a/x^2),x, algorithm="giac")
[Out]