Optimal. Leaf size=36 \[ \frac{\log (d+e x)}{e f-d g}-\frac{\log (f+g x)}{e f-d g} \]
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Rubi [A] time = 0.0145575, 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, Rules used = {616, 31} \[ \frac{\log (d+e x)}{e f-d g}-\frac{\log (f+g x)}{e f-d g} \]
Antiderivative was successfully verified.
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Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{d f+(e f+d g) x+e g x^2} \, dx &=-\frac{(e g) \int \frac{1}{e f+e g x} \, dx}{e f-d g}+\frac{(e g) \int \frac{1}{d g+e g x} \, dx}{e f-d g}\\ &=\frac{\log (d+e x)}{e f-d g}-\frac{\log (f+g x)}{e f-d g}\\ \end{align*}
Mathematica [A] time = 0.0103459, size = 26, normalized size = 0.72 \[ \frac{\log (d+e x)-\log (f+g x)}{e f-d g} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 37, normalized size = 1. \begin{align*} -{\frac{\ln \left ( ex+d \right ) }{dg-fe}}+{\frac{\ln \left ( gx+f \right ) }{dg-fe}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974656, size = 49, normalized size = 1.36 \begin{align*} \frac{\log \left (e x + d\right )}{e f - d g} - \frac{\log \left (g x + f\right )}{e f - d g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61521, size = 58, normalized size = 1.61 \begin{align*} \frac{\log \left (e x + d\right ) - \log \left (g x + f\right )}{e f - d g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.307197, size = 128, normalized size = 3.56 \begin{align*} \frac{\log{\left (x + \frac{- \frac{d^{2} g^{2}}{d g - e f} + \frac{2 d e f g}{d g - e f} + d g - \frac{e^{2} f^{2}}{d g - e f} + e f}{2 e g} \right )}}{d g - e f} - \frac{\log{\left (x + \frac{\frac{d^{2} g^{2}}{d g - e f} - \frac{2 d e f g}{d g - e f} + d g + \frac{e^{2} f^{2}}{d g - e f} + e f}{2 e g} \right )}}{d g - e f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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