Optimal. Leaf size=103 \[ -\frac{\sqrt{b x^2+c x^4} (4 b B-3 A c)}{8 b^2 x^3}+\frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{5/2}}-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5} \]
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Rubi [A] time = 0.165402, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2038, 2025, 2008, 206} \[ -\frac{\sqrt{b x^2+c x^4} (4 b B-3 A c)}{8 b^2 x^3}+\frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{5/2}}-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2025
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^4 \sqrt{b x^2+c x^4}} \, dx &=-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5}-\frac{(-4 b B+3 A c) \int \frac{1}{x^2 \sqrt{b x^2+c x^4}} \, dx}{4 b}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5}-\frac{(4 b B-3 A c) \sqrt{b x^2+c x^4}}{8 b^2 x^3}-\frac{(c (4 b B-3 A c)) \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx}{8 b^2}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5}-\frac{(4 b B-3 A c) \sqrt{b x^2+c x^4}}{8 b^2 x^3}+\frac{(c (4 b B-3 A c)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )}{8 b^2}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5}-\frac{(4 b B-3 A c) \sqrt{b x^2+c x^4}}{8 b^2 x^3}+\frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.149729, size = 104, normalized size = 1.01 \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )} \left (b \sqrt{\frac{c x^2}{b}+1} \left (2 A b-3 A c x^2+4 b B x^2\right )+c x^4 (3 A c-4 b B) \tanh ^{-1}\left (\sqrt{\frac{c x^2}{b}+1}\right )\right )}{8 b^3 x^5 \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 146, normalized size = 1.4 \begin{align*} -{\frac{1}{8\,{x}^{3}}\sqrt{c{x}^{2}+b} \left ( 3\,A\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}b{c}^{2}+4\,B{b}^{5/2}\sqrt{c{x}^{2}+b}{x}^{2}-4\,B\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}{b}^{2}c-3\,A{b}^{3/2}\sqrt{c{x}^{2}+b}{x}^{2}c+2\,A{b}^{5/2}\sqrt{c{x}^{2}+b} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{b}^{-{\frac{7}{2}}}} \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}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42975, size = 450, normalized size = 4.37 \begin{align*} \left [-\frac{{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt{b} x^{5} \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}}{\left (2 \, A b^{2} +{\left (4 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{16 \, b^{3} x^{5}}, -\frac{{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt{-b} x^{5} \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}}{\left (2 \, A b^{2} +{\left (4 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{8 \, b^{3} x^{5}}\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^{4} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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