Optimal. Leaf size=36 \[ \frac{\log (a+b x)}{b c-a d}-\frac{\log (c+d x)}{b c-a d} \]
[Out]
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Rubi [A] time = 0.0371094, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{\log (a+b x)}{b c-a d}-\frac{\log (c+d x)}{b c-a d} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 5.62547, size = 31, normalized size = 0.86 \[ - \frac{2 \operatorname{atanh}{\left (\frac{a d + b c + 2 b d x}{a d - b c} \right )}}{a d - b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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Mathematica [A] time = 0.020117, size = 26, normalized size = 0.72 \[ \frac{\log (a+b x)-\log (c+d x)}{b c-a d} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^(-1),x]
[Out]
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Maple [A] time = 0.008, size = 37, normalized size = 1. \[ -{\frac{\ln \left ( bx+a \right ) }{ad-bc}}+{\frac{\ln \left ( dx+c \right ) }{ad-bc}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a*c+(a*d+b*c)*x+x^2*b*d),x)
[Out]
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Maxima [A] time = 0.749696, size = 49, normalized size = 1.36 \[ \frac{\log \left (b x + a\right )}{b c - a d} - \frac{\log \left (d x + c\right )}{b c - a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203341, size = 35, normalized size = 0.97 \[ \frac{\log \left (b x + a\right ) - \log \left (d x + c\right )}{b c - a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.971554, size = 128, normalized size = 3.56 \[ \frac{\log{\left (x + \frac{- \frac{a^{2} d^{2}}{a d - b c} + \frac{2 a b c d}{a d - b c} + a d - \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} - \frac{\log{\left (x + \frac{\frac{a^{2} d^{2}}{a d - b c} - \frac{2 a b c d}{a d - b c} + a d + \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="giac")
[Out]