Optimal. Leaf size=62 \[ \frac{c \tanh ^{-1}\left (\frac{c+2 d x}{\sqrt{c^2-4 b d}}\right )}{b \sqrt{c^2-4 b d}}-\frac{\log \left (b+c x+d x^2\right )}{2 b}+\frac{\log (x)}{b} \]
[Out]
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Rubi [A] time = 0.112362, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ \frac{c \tanh ^{-1}\left (\frac{c+2 d x}{\sqrt{c^2-4 b d}}\right )}{b \sqrt{c^2-4 b d}}-\frac{\log \left (b+c x+d x^2\right )}{2 b}+\frac{\log (x)}{b} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2 + d*x^3)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 20.222, size = 54, normalized size = 0.87 \[ \frac{c \operatorname{atanh}{\left (\frac{c + 2 d x}{\sqrt{- 4 b d + c^{2}}} \right )}}{b \sqrt{- 4 b d + c^{2}}} + \frac{\log{\left (x \right )}}{b} - \frac{\log{\left (b + c x + d x^{2} \right )}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d*x**3+c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.131815, size = 61, normalized size = 0.98 \[ -\frac{\frac{2 c \tan ^{-1}\left (\frac{c+2 d x}{\sqrt{4 b d-c^2}}\right )}{\sqrt{4 b d-c^2}}+\log (b+x (c+d x))-2 \log (x)}{2 b} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2 + d*x^3)^(-1),x]
[Out]
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Maple [A] time = 0.013, size = 62, normalized size = 1. \[ -{\frac{\ln \left ( d{x}^{2}+cx+b \right ) }{2\,b}}-{\frac{c}{b}\arctan \left ({(2\,dx+c){\frac{1}{\sqrt{4\,bd-{c}^{2}}}}} \right ){\frac{1}{\sqrt{4\,bd-{c}^{2}}}}}+{\frac{\ln \left ( x \right ) }{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d*x^3+c*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/(d*x^3 + c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.304896, size = 1, normalized size = 0.02 \[ \left [\frac{c \log \left (\frac{c^{3} - 4 \, b c d + 2 \,{\left (c^{2} d - 4 \, b d^{2}\right )} x +{\left (2 \, d^{2} x^{2} + 2 \, c d x + c^{2} - 2 \, b d\right )} \sqrt{c^{2} - 4 \, b d}}{d x^{2} + c x + b}\right ) - \sqrt{c^{2} - 4 \, b d}{\left (\log \left (d x^{2} + c x + b\right ) - 2 \, \log \left (x\right )\right )}}{2 \, \sqrt{c^{2} - 4 \, b d} b}, -\frac{2 \, c \arctan \left (-\frac{\sqrt{-c^{2} + 4 \, b d}{\left (2 \, d x + c\right )}}{c^{2} - 4 \, b d}\right ) + \sqrt{-c^{2} + 4 \, b d}{\left (\log \left (d x^{2} + c x + b\right ) - 2 \, \log \left (x\right )\right )}}{2 \, \sqrt{-c^{2} + 4 \, b d} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d*x^3 + c*x^2 + b*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.67278, size = 564, normalized size = 9.1 \[ \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) \log{\left (x + \frac{24 b^{4} d^{2} \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} - 14 b^{3} c^{2} d \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} - 12 b^{3} d^{2} \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) + 2 b^{2} c^{4} \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} + 3 b^{2} c^{2} d \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) - 12 b^{2} d^{2} + 11 b c^{2} d - 2 c^{4}}{9 b c d^{2} - 2 c^{3} d} \right )} + \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) \log{\left (x + \frac{24 b^{4} d^{2} \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} - 14 b^{3} c^{2} d \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} - 12 b^{3} d^{2} \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) + 2 b^{2} c^{4} \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} + 3 b^{2} c^{2} d \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) - 12 b^{2} d^{2} + 11 b c^{2} d - 2 c^{4}}{9 b c d^{2} - 2 c^{3} d} \right )} + \frac{\log{\left (x \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d*x**3+c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.263704, size = 84, normalized size = 1.35 \[ -\frac{c \arctan \left (\frac{2 \, d x + c}{\sqrt{-c^{2} + 4 \, b d}}\right )}{\sqrt{-c^{2} + 4 \, b d} b} - \frac{{\rm ln}\left (d x^{2} + c x + b\right )}{2 \, b} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d*x^3 + c*x^2 + b*x),x, algorithm="giac")
[Out]