Optimal. Leaf size=42 \[ -\frac{A \log (a+b x)}{a^2}+\frac{A \log (x)}{a^2}+\frac{A b-a B}{a b (a+b x)} \]
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Rubi [A] time = 0.0249285, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ -\frac{A \log (a+b x)}{a^2}+\frac{A \log (x)}{a^2}+\frac{A b-a B}{a b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x (a+b x)^2} \, dx &=\int \left (\frac{A}{a^2 x}+\frac{-A b+a B}{a (a+b x)^2}-\frac{A b}{a^2 (a+b x)}\right ) \, dx\\ &=\frac{A b-a B}{a b (a+b x)}+\frac{A \log (x)}{a^2}-\frac{A \log (a+b x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0259676, size = 38, normalized size = 0.9 \[ \frac{\frac{a (A b-a B)}{b (a+b x)}-A \log (a+b x)+A \log (x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 46, normalized size = 1.1 \begin{align*}{\frac{A\ln \left ( x \right ) }{{a}^{2}}}+{\frac{A}{a \left ( bx+a \right ) }}-{\frac{B}{b \left ( bx+a \right ) }}-{\frac{A\ln \left ( bx+a \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03312, size = 59, normalized size = 1.4 \begin{align*} -\frac{B a - A b}{a b^{2} x + a^{2} b} - \frac{A \log \left (b x + a\right )}{a^{2}} + \frac{A \log \left (x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96464, size = 132, normalized size = 3.14 \begin{align*} -\frac{B a^{2} - A a b +{\left (A b^{2} x + A a b\right )} \log \left (b x + a\right ) -{\left (A b^{2} x + A a b\right )} \log \left (x\right )}{a^{2} b^{2} x + a^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.618404, size = 32, normalized size = 0.76 \begin{align*} \frac{A \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{2}} - \frac{- A b + B a}{a^{2} b + a b^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16804, size = 74, normalized size = 1.76 \begin{align*} b{\left (\frac{A \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{2} b} - \frac{\frac{B a}{b x + a} - \frac{A b}{b x + a}}{a b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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