Optimal. Leaf size=23 \[ \frac{b}{a^2 (a x+b)}+\frac{\log (a x+b)}{a^2} \]
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Rubi [A] time = 0.0147991, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac{b}{a^2 (a x+b)}+\frac{\log (a x+b)}{a^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^2 x} \, dx &=\int \frac{x}{(b+a x)^2} \, dx\\ &=\int \left (-\frac{b}{a (b+a x)^2}+\frac{1}{a (b+a x)}\right ) \, dx\\ &=\frac{b}{a^2 (b+a x)}+\frac{\log (b+a x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0060767, size = 20, normalized size = 0.87 \[ \frac{\frac{b}{a x+b}+\log (a x+b)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 24, normalized size = 1. \begin{align*}{\frac{b}{{a}^{2} \left ( ax+b \right ) }}+{\frac{\ln \left ( ax+b \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01971, size = 35, normalized size = 1.52 \begin{align*} \frac{b}{a^{3} x + a^{2} b} + \frac{\log \left (a x + b\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41431, size = 62, normalized size = 2.7 \begin{align*} \frac{{\left (a x + b\right )} \log \left (a x + b\right ) + b}{a^{3} x + a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.277199, size = 20, normalized size = 0.87 \begin{align*} \frac{b}{a^{3} x + a^{2} b} + \frac{\log{\left (a x + b \right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09878, size = 32, normalized size = 1.39 \begin{align*} \frac{\log \left ({\left | a x + b \right |}\right )}{a^{2}} + \frac{b}{{\left (a x + b\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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