Optimal. Leaf size=62 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x+c x^2\right )}{2 a}+\frac{\log (x)}{a} \]
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Rubi [A] time = 0.106115, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x+c x^2\right )}{2 a}+\frac{\log (x)}{a} \]
Antiderivative was successfully verified.
[In] Int[1/((c + a/x^2 + b/x)*x^3),x]
[Out]
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Rubi in Sympy [A] time = 24.8998, size = 54, normalized size = 0.87 \[ \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (a + b x + c x^{2} \right )}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c+a/x**2+b/x)/x**3,x)
[Out]
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Mathematica [A] time = 0.112243, size = 61, normalized size = 0.98 \[ -\frac{\frac{2 b \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\log (a+x (b+c x))-2 \log (x)}{2 a} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c + a/x^2 + b/x)*x^3),x]
[Out]
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Maple [A] time = 0.007, size = 62, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{2\,a}}-{\frac{b}{a}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c+a/x^2+b/x)/x^3,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^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.282766, size = 1, normalized size = 0.02 \[ \left [\frac{b \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (\log \left (c x^{2} + b x + a\right ) - 2 \, \log \left (x\right )\right )}}{2 \, \sqrt{b^{2} - 4 \, a c} a}, -\frac{2 \, b \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (\log \left (c x^{2} + b x + a\right ) - 2 \, \log \left (x\right )\right )}}{2 \, \sqrt{-b^{2} + 4 \, a c} a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c + b/x + a/x^2)*x^3),x, algorithm="fricas")
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Sympy [A] time = 6.62289, size = 564, normalized size = 9.1 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) \log{\left (x + \frac{24 a^{4} c^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) + 2 a^{2} b^{4} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) \log{\left (x + \frac{24 a^{4} c^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) + 2 a^{2} b^{4} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \frac{\log{\left (x \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c+a/x**2+b/x)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.305839, size = 84, normalized size = 1.35 \[ -\frac{b \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a} - \frac{{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c + b/x + a/x^2)*x^3),x, algorithm="giac")
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