Optimal. Leaf size=23 \[ \frac{a}{b^2 (a+b x)}+\frac{\log (a+b x)}{b^2} \]
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Rubi [A] time = 0.0119928, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{a}{b^2 (a+b x)}+\frac{\log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x}{(a+b x)^2} \, dx &=\int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx\\ &=\frac{a}{b^2 (a+b x)}+\frac{\log (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0105572, size = 20, normalized size = 0.87 \[ \frac{\frac{a}{a+b x}+\log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 24, normalized size = 1. \begin{align*}{\frac{a}{{b}^{2} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02621, size = 35, normalized size = 1.52 \begin{align*} \frac{a}{b^{3} x + a b^{2}} + \frac{\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 = 1.50717, size = 62, normalized size = 2.7 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a\right ) + a}{b^{3} x + a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.399283, size = 20, normalized size = 0.87 \begin{align*} \frac{a}{a b^{2} + b^{3} x} + \frac{\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 = 1.17541, size = 57, normalized size = 2.48 \begin{align*} -\frac{\frac{\log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x + a\right )} b}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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