Optimal. Leaf size=103 \[ -\frac{c (4 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{3/2}}-\frac{\sqrt{b x^2+c x^4} (4 b B-A c)}{8 b x^3}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{4 b x^7} \]
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Rubi [A] time = 0.157621, 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, 2020, 2008, 206} \[ -\frac{c (4 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{3/2}}-\frac{\sqrt{b x^2+c x^4} (4 b B-A c)}{8 b x^3}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{4 b x^7} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2020
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{x^6} \, dx &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{4 b x^7}-\frac{(-4 b B+A c) \int \frac{\sqrt{b x^2+c x^4}}{x^4} \, dx}{4 b}\\ &=-\frac{(4 b B-A c) \sqrt{b x^2+c x^4}}{8 b x^3}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{4 b x^7}+\frac{(c (4 b B-A c)) \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx}{8 b}\\ &=-\frac{(4 b B-A c) \sqrt{b x^2+c x^4}}{8 b x^3}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{4 b x^7}-\frac{(c (4 b B-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}\\ &=-\frac{(4 b B-A c) \sqrt{b x^2+c x^4}}{8 b x^3}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{4 b x^7}-\frac{c (4 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.107429, size = 95, normalized size = 0.92 \[ -\frac{\left (b+c x^2\right ) \left (2 A b+A c x^2+4 b B x^2\right )+c x^4 \sqrt{\frac{c x^2}{b}+1} (4 b B-A c) \tanh ^{-1}\left (\sqrt{\frac{c x^2}{b}+1}\right )}{8 b x^3 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 174, normalized size = 1.7 \begin{align*}{\frac{1}{8\,{x}^{5}{b}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( A\sqrt{b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}{c}^{2}-4\,B{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}c-A\sqrt{c{x}^{2}+b}{x}^{4}{c}^{2}+4\,B\sqrt{c{x}^{2}+b}{x}^{4}bc+A \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{2}c-4\,B \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{2}b-2\,A \left ( c{x}^{2}+b \right ) ^{3/2}b \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}} \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{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07663, size = 437, normalized size = 4.24 \begin{align*} \left [-\frac{{\left (4 \, B b c - 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} + A b c\right )} x^{2}\right )}}{16 \, b^{2} x^{5}}, \frac{{\left (4 \, B b c - 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} + A b c\right )} x^{2}\right )}}{8 \, b^{2} 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{\sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24954, size = 178, normalized size = 1.73 \begin{align*} \frac{\frac{{\left (4 \, B b c^{2} \mathrm{sgn}\left (x\right ) - A c^{3} \mathrm{sgn}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{4 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} B b c^{2} \mathrm{sgn}\left (x\right ) - 4 \, \sqrt{c x^{2} + b} B b^{2} c^{2} \mathrm{sgn}\left (x\right ) +{\left (c x^{2} + b\right )}^{\frac{3}{2}} A c^{3} \mathrm{sgn}\left (x\right ) + \sqrt{c x^{2} + b} A b c^{3} \mathrm{sgn}\left (x\right )}{b c^{2} x^{4}}}{8 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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