Optimal. Leaf size=28 \[ -\frac{b \log (x)}{a^2}+\frac{b \log (a+b x)}{a^2}-\frac{1}{a x} \]
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Rubi [A] time = 0.012942, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {44} \[ -\frac{b \log (x)}{a^2}+\frac{b \log (a+b x)}{a^2}-\frac{1}{a x} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^2 (a+b x)} \, dx &=\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx\\ &=-\frac{1}{a x}-\frac{b \log (x)}{a^2}+\frac{b \log (a+b x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0041583, size = 28, normalized size = 1. \[ -\frac{b \log (x)}{a^2}+\frac{b \log (a+b x)}{a^2}-\frac{1}{a x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 29, normalized size = 1. \begin{align*} -{\frac{1}{ax}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}}+{\frac{b\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 = 0.958913, size = 38, normalized size = 1.36 \begin{align*} \frac{b \log \left (b x + a\right )}{a^{2}} - \frac{b \log \left (x\right )}{a^{2}} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99516, size = 61, normalized size = 2.18 \begin{align*} \frac{b x \log \left (b x + a\right ) - b x \log \left (x\right ) - a}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.325009, size = 19, normalized size = 0.68 \begin{align*} - \frac{1}{a x} + \frac{b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07897, size = 41, normalized size = 1.46 \begin{align*} \frac{b \log \left ({\left | b x + a \right |}\right )}{a^{2}} - \frac{b \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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