Optimal. Leaf size=34 \[ \frac{A \log (x)}{a}-\frac{(A b-a B) \log \left (a+b x^2\right )}{2 a b} \]
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Rubi [A] time = 0.0335423, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 72} \[ \frac{A \log (x)}{a}-\frac{(A b-a B) \log \left (a+b x^2\right )}{2 a b} \]
Antiderivative was successfully verified.
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Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x \left (a+b x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x (a+b x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A}{a x}+\frac{-A b+a B}{a (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{A \log (x)}{a}-\frac{(A b-a B) \log \left (a+b x^2\right )}{2 a b}\\ \end{align*}
Mathematica [A] time = 0.0121514, size = 34, normalized size = 1. \[ \frac{(a B-A b) \log \left (a+b x^2\right )}{2 a b}+\frac{A \log (x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 37, normalized size = 1.1 \begin{align*}{\frac{A\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,a}}+{\frac{\ln \left ( b{x}^{2}+a \right ) B}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00515, size = 47, normalized size = 1.38 \begin{align*} \frac{A \log \left (x^{2}\right )}{2 \, a} + \frac{{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21652, size = 74, normalized size = 2.18 \begin{align*} \frac{2 \, A b \log \left (x\right ) +{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.654914, size = 26, normalized size = 0.76 \begin{align*} \frac{A \log{\left (x \right )}}{a} + \frac{\left (- A b + B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14665, size = 49, normalized size = 1.44 \begin{align*} \frac{A \log \left (x^{2}\right )}{2 \, a} + \frac{{\left (B a - A b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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