Optimal. Leaf size=45 \[ \frac{x (b c-a d)}{b^2}-\frac{a (b c-a d) \log (a+b x)}{b^3}+\frac{d x^2}{2 b} \]
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Rubi [A] time = 0.0323585, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {77} \[ \frac{x (b c-a d)}{b^2}-\frac{a (b c-a d) \log (a+b x)}{b^3}+\frac{d x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x (c+d x)}{a+b x} \, dx &=\int \left (\frac{b c-a d}{b^2}+\frac{d x}{b}+\frac{a (-b c+a d)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{(b c-a d) x}{b^2}+\frac{d x^2}{2 b}-\frac{a (b c-a d) \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0127889, size = 41, normalized size = 0.91 \[ \frac{b x (-2 a d+2 b c+b d x)+2 a (a d-b c) \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 52, normalized size = 1.2 \begin{align*}{\frac{d{x}^{2}}{2\,b}}-{\frac{adx}{{b}^{2}}}+{\frac{cx}{b}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) d}{{b}^{3}}}-{\frac{a\ln \left ( bx+a \right ) c}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08073, size = 62, normalized size = 1.38 \begin{align*} \frac{b d x^{2} + 2 \,{\left (b c - a d\right )} x}{2 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89706, size = 103, normalized size = 2.29 \begin{align*} \frac{b^{2} d x^{2} + 2 \,{\left (b^{2} c - a b d\right )} x - 2 \,{\left (a b c - a^{2} d\right )} \log \left (b x + a\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.388438, size = 37, normalized size = 0.82 \begin{align*} \frac{a \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{3}} + \frac{d x^{2}}{2 b} - \frac{x \left (a d - b c\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23569, size = 62, normalized size = 1.38 \begin{align*} \frac{b d x^{2} + 2 \, b c x - 2 \, a d x}{2 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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