Optimal. Leaf size=66 \[ -\frac{\left (b+2 c x^2\right ) (3 b B-4 A c)}{3 b^3 \sqrt{b x^2+c x^4}}-\frac{A}{3 b x^2 \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.163637, antiderivative size = 66, 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, 613} \[ -\frac{\left (b+2 c x^2\right ) (3 b B-4 A c)}{3 b^3 \sqrt{b x^2+c x^4}}-\frac{A}{3 b x^2 \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 792
Rule 613
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x \left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x \left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{A}{3 b x^2 \sqrt{b x^2+c x^4}}+\frac{\left (b B-A c+\frac{1}{2} (b B-2 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )}{3 b}\\ &=-\frac{A}{3 b x^2 \sqrt{b x^2+c x^4}}-\frac{(3 b B-4 A c) \left (b+2 c x^2\right )}{3 b^3 \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.0224842, size = 64, normalized size = 0.97 \[ \frac{A \left (-b^2+4 b c x^2+8 c^2 x^4\right )-3 b B x^2 \left (b+2 c x^2\right )}{3 b^3 x^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 66, normalized size = 1. \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -8\,A{c}^{2}{x}^{4}+6\,B{x}^{4}bc-4\,Abc{x}^{2}+3\,B{x}^{2}{b}^{2}+A{b}^{2} \right ) }{3\,{b}^{3}} \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 [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 = 1.25112, size = 149, normalized size = 2.26 \begin{align*} -\frac{{\left (2 \,{\left (3 \, B b c - 4 \, A c^{2}\right )} x^{4} + A b^{2} +{\left (3 \, B b^{2} - 4 \, A b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3 \,{\left (b^{3} c x^{6} + b^{4} x^{4}\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}}{x \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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