Optimal. Leaf size=30 \[ \frac{d \log (x)}{b}-\frac{(c d-b e) \log (b+c x)}{b c} \]
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Rubi [A] time = 0.0200777, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {631} \[ \frac{d \log (x)}{b}-\frac{(c d-b e) \log (b+c x)}{b c} \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int \frac{d+e x}{b x+c x^2} \, dx &=\int \left (\frac{d}{b x}+\frac{-c d+b e}{b (b+c x)}\right ) \, dx\\ &=\frac{d \log (x)}{b}-\frac{(c d-b e) \log (b+c x)}{b c}\\ \end{align*}
Mathematica [A] time = 0.008958, size = 29, normalized size = 0.97 \[ \frac{(b e-c d) \log (b+c x)}{b c}+\frac{d \log (x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 32, normalized size = 1.1 \begin{align*}{\frac{d\ln \left ( x \right ) }{b}}+{\frac{\ln \left ( cx+b \right ) e}{c}}-{\frac{\ln \left ( cx+b \right ) d}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10816, size = 41, normalized size = 1.37 \begin{align*} \frac{d \log \left (x\right )}{b} - \frac{{\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82405, size = 63, normalized size = 2.1 \begin{align*} \frac{c d \log \left (x\right ) -{\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.575087, size = 41, normalized size = 1.37 \begin{align*} \frac{d \log{\left (x \right )}}{b} + \frac{\left (b e - c d\right ) \log{\left (x + \frac{- b d + \frac{b \left (b e - c d\right )}{c}}{b e - 2 c d} \right )}}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15538, size = 45, normalized size = 1.5 \begin{align*} \frac{d \log \left ({\left | x \right |}\right )}{b} - \frac{{\left (c d - b e\right )} \log \left ({\left | c x + b \right |}\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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