Optimal. Leaf size=42 \[ \frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]
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Rubi [A] time = 0.0318716, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1593, 453, 205} \[ \frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 453
Rule 205
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) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0249807, size = 42, normalized size = 1. \[ \frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 48, normalized size = 1.1 \begin{align*} -{\frac{A}{bx}}-{\frac{Ac}{b}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{B\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \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 = 0.764951, size = 228, normalized size = 5.43 \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 = 0.452095, size = 82, normalized size = 1.95 \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 = 1.24151, size = 49, normalized size = 1.17 \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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