Optimal. Leaf size=25 \[ \frac{(A b-a B) \log (a+b x)}{b^2}+\frac{B x}{b} \]
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Rubi [A] time = 0.0179, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{(A b-a B) \log (a+b x)}{b^2}+\frac{B x}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{A+B x}{a+b x} \, dx &=\int \left (\frac{B}{b}+\frac{A b-a B}{b (a+b x)}\right ) \, dx\\ &=\frac{B x}{b}+\frac{(A b-a B) \log (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0070388, size = 25, normalized size = 1. \[ \frac{(A b-a B) \log (a+b x)}{b^2}+\frac{B x}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 32, normalized size = 1.3 \begin{align*}{\frac{Bx}{b}}+{\frac{\ln \left ( bx+a \right ) A}{b}}-{\frac{\ln \left ( bx+a \right ) Ba}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24185, size = 35, normalized size = 1.4 \begin{align*} \frac{B x}{b} - \frac{{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.766, size = 54, normalized size = 2.16 \begin{align*} \frac{B b x -{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.288708, size = 20, normalized size = 0.8 \begin{align*} \frac{B x}{b} - \frac{\left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.80959, size = 36, normalized size = 1.44 \begin{align*} \frac{B x}{b} - \frac{{\left (B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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