Optimal. Leaf size=15 \[ \frac{\log \left (a x^2+b\right )}{2 a} \]
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Rubi [A] time = 0.004795, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 260} \[ \frac{\log \left (a x^2+b\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{\frac{b}{x}+a x} \, dx &=\int \frac{x}{b+a x^2} \, dx\\ &=\frac{\log \left (b+a x^2\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0026931, size = 15, normalized size = 1. \[ \frac{\log \left (a x^2+b\right )}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 14, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26634, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (a x^{2} + b\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.651248, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (a x^{2} + b\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.1141, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (a x^{2} + b \right )}}{2 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12146, size = 19, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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