Optimal. Leaf size=25 \[ \frac{(c d-b e) \log (b+c x)}{c^2}+\frac{e x}{c} \]
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Rubi [A] time = 0.0230134, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {765} \[ \frac{(c d-b e) \log (b+c x)}{c^2}+\frac{e x}{c} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{x (d+e x)}{b x+c x^2} \, dx &=\int \left (\frac{e}{c}+\frac{c d-b e}{c (b+c x)}\right ) \, dx\\ &=\frac{e x}{c}+\frac{(c d-b e) \log (b+c x)}{c^2}\\ \end{align*}
Mathematica [A] time = 0.0071542, size = 25, normalized size = 1. \[ \frac{(c d-b e) \log (b+c x)}{c^2}+\frac{e x}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 1.3 \begin{align*}{\frac{ex}{c}}-{\frac{\ln \left ( cx+b \right ) be}{{c}^{2}}}+{\frac{\ln \left ( cx+b \right ) d}{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16445, size = 34, normalized size = 1.36 \begin{align*} \frac{e x}{c} + \frac{{\left (c d - b e\right )} \log \left (c x + b\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90932, size = 54, normalized size = 2.16 \begin{align*} \frac{c e x +{\left (c d - b e\right )} \log \left (c x + b\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.492257, size = 20, normalized size = 0.8 \begin{align*} \frac{e x}{c} - \frac{\left (b e - c d\right ) \log{\left (b + c x \right )}}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16975, size = 38, normalized size = 1.52 \begin{align*} \frac{x e}{c} + \frac{{\left (c d - b e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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