Optimal. Leaf size=41 \[ \frac{(A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]
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Rubi [A] time = 0.0344379, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1593, 453, 208} \[ \frac{(A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 453
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x^2}{b x^2-c x^4} \, dx &=\int \frac{A+B x^2}{x^2 \left (b-c x^2\right )} \, dx\\ &=-\frac{A}{b x}+\frac{(b B+A c) \int \frac{1}{b-c x^2} \, dx}{b}\\ &=-\frac{A}{b x}+\frac{(b B+A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.023628, size = 41, normalized size = 1. \[ \frac{(A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 39, normalized size = 1. \begin{align*} -{\frac{-Ac-Bb}{b}{\it Artanh} \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{A}{bx}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.747638, size = 230, normalized size = 5.61 \begin{align*} \left [\frac{{\left (B b + A c\right )} \sqrt{b c} x \log \left (\frac{c x^{2} + 2 \, \sqrt{b c} x + b}{c x^{2} - b}\right ) - 2 \, A b c}{2 \, b^{2} c x}, -\frac{{\left (B b + A c\right )} \sqrt{-b c} x \arctan \left (\frac{\sqrt{-b c} x}{b}\right ) + A b c}{b^{2} c x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.467934, size = 75, normalized size = 1.83 \begin{align*} - \frac{A}{b x} - \frac{\sqrt{\frac{1}{b^{3} c}} \left (A c + B b\right ) \log{\left (- b^{2} \sqrt{\frac{1}{b^{3} c}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{b^{3} c}} \left (A c + B b\right ) \log{\left (b^{2} \sqrt{\frac{1}{b^{3} c}} + x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25921, size = 51, normalized size = 1.24 \begin{align*} -\frac{{\left (B b + A c\right )} \arctan \left (\frac{c x}{\sqrt{-b c}}\right )}{\sqrt{-b c} b} - \frac{A}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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