Optimal. Leaf size=45 \[ -\frac{x (c d-b e)}{e^2}+\frac{d (c d-b e) \log (d+e x)}{e^3}+\frac{c x^2}{2 e} \]
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Rubi [A] time = 0.0340222, antiderivative size = 45, 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 = {698} \[ -\frac{x (c d-b e)}{e^2}+\frac{d (c d-b e) \log (d+e x)}{e^3}+\frac{c x^2}{2 e} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{b x+c x^2}{d+e x} \, dx &=\int \left (\frac{-c d+b e}{e^2}+\frac{c x}{e}+\frac{d (c d-b e)}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{(c d-b e) x}{e^2}+\frac{c x^2}{2 e}+\frac{d (c d-b e) \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0146041, size = 41, normalized size = 0.91 \[ \frac{e x (2 b e-2 c d+c e x)+2 d (c d-b e) \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 52, normalized size = 1.2 \begin{align*}{\frac{c{x}^{2}}{2\,e}}+{\frac{bx}{e}}-{\frac{cdx}{{e}^{2}}}-{\frac{d\ln \left ( ex+d \right ) b}{{e}^{2}}}+{\frac{{d}^{2}\ln \left ( ex+d \right ) c}{{e}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11148, size = 61, normalized size = 1.36 \begin{align*} \frac{c e x^{2} - 2 \,{\left (c d - b e\right )} x}{2 \, e^{2}} + \frac{{\left (c d^{2} - b d e\right )} \log \left (e x + d\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52674, size = 103, normalized size = 2.29 \begin{align*} \frac{c e^{2} x^{2} - 2 \,{\left (c d e - b e^{2}\right )} x + 2 \,{\left (c d^{2} - b d e\right )} \log \left (e x + d\right )}{2 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.24481, size = 37, normalized size = 0.82 \begin{align*} \frac{c x^{2}}{2 e} - \frac{d \left (b e - c d\right ) \log{\left (d + e x \right )}}{e^{3}} + \frac{x \left (b e - c d\right )}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38182, size = 63, normalized size = 1.4 \begin{align*}{\left (c d^{2} - b d e\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (c x^{2} e - 2 \, c d x + 2 \, b x e\right )} e^{\left (-2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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