Optimal. Leaf size=61 \[ -\frac{\sqrt{b x^2+c x^4} (3 b B-2 A c)}{3 b^2 x^2}-\frac{A \sqrt{b x^2+c x^4}}{3 b x^4} \]
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Rubi [A] time = 0.1684, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2034, 792, 650} \[ -\frac{\sqrt{b x^2+c x^4} (3 b B-2 A c)}{3 b^2 x^2}-\frac{A \sqrt{b x^2+c x^4}}{3 b x^4} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 792
Rule 650
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^3 \sqrt{b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 \sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac{A \sqrt{b x^2+c x^4}}{3 b x^4}+\frac{\left (-2 (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{3 b}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{3 b x^4}-\frac{(3 b B-2 A c) \sqrt{b x^2+c x^4}}{3 b^2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0238092, size = 43, normalized size = 0.7 \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )} \left (A \left (b-2 c x^2\right )+3 b B x^2\right )}{3 b^2 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 47, normalized size = 0.8 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -2\,A{x}^{2}c+3\,B{x}^{2}b+Ab \right ) }{3\,{b}^{2}{x}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{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 = 1.07458, size = 86, normalized size = 1.41 \begin{align*} -\frac{\sqrt{c x^{4} + b x^{2}}{\left ({\left (3 \, B b - 2 \, A c\right )} x^{2} + A b\right )}}{3 \, b^{2} x^{4}} \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}}{x^{3} \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 = 1.16651, size = 58, normalized size = 0.95 \begin{align*} -\frac{3 \, B b \sqrt{c + \frac{b}{x^{2}}} + A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} - 3 \, A \sqrt{c + \frac{b}{x^{2}}} c}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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