Optimal. Leaf size=80 \[ -\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 b x^6}-\frac{B \sqrt{b x^2+c x^4}}{x^2}+B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right ) \]
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Rubi [A] time = 0.195615, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2034, 792, 662, 620, 206} \[ -\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 b x^6}-\frac{B \sqrt{b x^2+c x^4}}{x^2}+B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right ) \]
Antiderivative was successfully verified.
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Rule 2034
Rule 792
Rule 662
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) \sqrt{b x+c x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 b x^6}+\frac{1}{2} B \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{B \sqrt{b x^2+c x^4}}{x^2}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 b x^6}+\frac{1}{2} (B c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac{B \sqrt{b x^2+c x^4}}{x^2}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 b x^6}+(B c) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )\\ &=-\frac{B \sqrt{b x^2+c x^4}}{x^2}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 b x^6}+B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.122246, size = 86, normalized size = 1.08 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (-A \left (b+c x^2\right )+\frac{3 \sqrt{b} B \sqrt{c} x^3 \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{\frac{c x^2}{b}+1}}-3 b B x^2\right )}{3 b x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 109, normalized size = 1.4 \begin{align*} -{\frac{1}{3\,b{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( -3\,B{c}^{3/2}\sqrt{c{x}^{2}+b}{x}^{4}+3\,B\sqrt{c} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{2}-3\,B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){x}^{3}bc+A\sqrt{c} \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}{\frac{1}{\sqrt{c}}}} \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 = 1.09415, size = 363, normalized size = 4.54 \begin{align*} \left [\frac{3 \, B b \sqrt{c} x^{4} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}}{\left ({\left (3 \, B b + A c\right )} x^{2} + A b\right )}}{6 \, b x^{4}}, -\frac{3 \, B b \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) + \sqrt{c x^{4} + b x^{2}}{\left ({\left (3 \, B b + A c\right )} x^{2} + A b\right )}}{3 \, b 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{\sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.54875, size = 220, normalized size = 2.75 \begin{align*} -\frac{1}{2} \, B \sqrt{c} \log \left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2}\right ) \mathrm{sgn}\left (x\right ) + \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} B b \sqrt{c} \mathrm{sgn}\left (x\right ) + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} A c^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) - 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} B b^{2} \sqrt{c} \mathrm{sgn}\left (x\right ) + 3 \, B b^{3} \sqrt{c} \mathrm{sgn}\left (x\right ) + A b^{2} c^{\frac{3}{2}} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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