Optimal. Leaf size=66 \[ \frac{B \sqrt{b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}} \]
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Rubi [A] time = 0.123152, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2034, 640, 620, 206} \[ \frac{B \sqrt{b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x \left (A+B x^2\right )}{\sqrt{b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac{B \sqrt{b x^2+c x^4}}{2 c}+\frac{(-b B+2 A c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{4 c}\\ &=\frac{B \sqrt{b x^2+c x^4}}{2 c}+\frac{(-b B+2 A c) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c}\\ &=\frac{B \sqrt{b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0434992, size = 81, normalized size = 1.23 \[ \frac{x \left (B \sqrt{c} x \left (b+c x^2\right )-\sqrt{b+c x^2} (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right )\right )}{2 c^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 88, normalized size = 1.3 \begin{align*}{\frac{x}{2}\sqrt{c{x}^{2}+b} \left ( B{c}^{{\frac{3}{2}}}\sqrt{c{x}^{2}+b}x+2\,A\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){c}^{2}-B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) bc \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{c}^{-{\frac{5}{2}}}} \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 = 1.1539, size = 302, normalized size = 4.58 \begin{align*} \left [\frac{2 \, \sqrt{c x^{4} + b x^{2}} B c -{\left (B b - 2 \, A c\right )} \sqrt{c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right )}{4 \, c^{2}}, \frac{\sqrt{c x^{4} + b x^{2}} B c +{\left (B b - 2 \, A c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right )}{2 \, c^{2}}\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{x \left (A + B x^{2}\right )}{\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 = 1.23461, size = 90, normalized size = 1.36 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}} B}{2 \, c} + \frac{{\left (B b - 2 \, A c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2}}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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