Optimal. Leaf size=37 \[ -\frac{A b-x^2 (b B-2 A c)}{b^2 \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.121296, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2034, 636} \[ -\frac{A b-x^2 (b B-2 A c)}{b^2 \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 636
Rubi steps
\begin{align*} \int \frac{x \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{A b-(b B-2 A c) x^2}{b^2 \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.0187079, size = 37, normalized size = 1. \[ \frac{b B x^2-A \left (b+2 c x^2\right )}{b^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 47, normalized size = 1.3 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ){x}^{2} \left ( 2\,A{x}^{2}c-B{x}^{2}b+Ab \right ) }{{b}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1736, size = 88, normalized size = 2.38 \begin{align*} -A{\left (\frac{2 \, c x^{2}}{\sqrt{c x^{4} + b x^{2}} b^{2}} + \frac{1}{\sqrt{c x^{4} + b x^{2}} b}\right )} + \frac{B x^{2}}{\sqrt{c x^{4} + b x^{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32803, size = 93, normalized size = 2.51 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}{\left ({\left (B b - 2 \, A c\right )} x^{2} - A b\right )}}{b^{2} c x^{4} + b^{3} x^{2}} \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 )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18676, size = 49, normalized size = 1.32 \begin{align*} \frac{\frac{{\left (B b - 2 \, A c\right )} x^{2}}{b^{2}} - \frac{A}{b}}{\sqrt{c x^{4} + b x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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