Optimal. Leaf size=55 \[ \frac{B \sqrt{b x^2+c x^4}}{c x}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0197414, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1145, 2008, 206} \[ \frac{B \sqrt{b x^2+c x^4}}{c x}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 1145
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x^2}{\sqrt{b x^2+c x^4}} \, dx &=\frac{B \sqrt{b x^2+c x^4}}{c x}+A \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{B \sqrt{b x^2+c x^4}}{c x}-A \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )\\ &=\frac{B \sqrt{b x^2+c x^4}}{c x}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0319559, size = 73, normalized size = 1.33 \[ \frac{x \left (\sqrt{b} B \left (b+c x^2\right )-A c \sqrt{b+c x^2} \tanh ^{-1}\left (\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right )\right )}{\sqrt{b} c \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 72, normalized size = 1.3 \begin{align*} -{\frac{x}{c}\sqrt{c{x}^{2}+b} \left ( A\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) c-B\sqrt{c{x}^{2}+b}\sqrt{b} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{\frac{1}{\sqrt{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06763, size = 300, normalized size = 5.45 \begin{align*} \left [\frac{A \sqrt{b} c x \log \left (-\frac{c x^{3} + 2 \, b x - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}} B b}{2 \, b c x}, \frac{A \sqrt{-b} c x \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}{c x^{3} + b x}\right ) + \sqrt{c x^{4} + b x^{2}} B b}{b c x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x^{2}}{\sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19273, size = 81, normalized size = 1.47 \begin{align*} \frac{A \log \left ({\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2}\right )}{2 \, \sqrt{b}} - \frac{2 \, B \sqrt{b}}{{\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2} - c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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