Optimal. Leaf size=97 \[ \frac{\sqrt{b x^2+c x^4} (2 A c+b B)}{2 b}+\frac{(2 A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 \sqrt{c}}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{b x^4} \]
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Rubi [A] time = 0.213395, antiderivative size = 97, 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, 664, 620, 206} \[ \frac{\sqrt{b x^2+c x^4} (2 A c+b B)}{2 b}+\frac{(2 A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 \sqrt{c}}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{b x^4} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 792
Rule 664
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) \sqrt{b x+c x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{b x^4}+\frac{\left (-2 (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x} \, dx,x,x^2\right )}{b}\\ &=\frac{(b B+2 A c) \sqrt{b x^2+c x^4}}{2 b}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{b x^4}+\frac{1}{4} (b B+2 A c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac{(b B+2 A c) \sqrt{b x^2+c x^4}}{2 b}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{b x^4}+\frac{1}{2} (b B+2 A 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 B+2 A c) \sqrt{b x^2+c x^4}}{2 b}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{b x^4}+\frac{(b B+2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.142748, size = 78, normalized size = 0.8 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\frac{x (2 A c+b B) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{c} \sqrt{\frac{c x^2}{b}+1}}-2 A+B x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 130, normalized size = 1.3 \begin{align*}{\frac{1}{2\,b{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 2\,A{c}^{3/2}\sqrt{c{x}^{2}+b}{x}^{2}+B\sqrt{c}\sqrt{c{x}^{2}+b}{x}^{2}b-2\,A\sqrt{c} \left ( c{x}^{2}+b \right ) ^{3/2}+2\,A\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) xbc+B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) x{b}^{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 = 1.18882, size = 363, normalized size = 3.74 \begin{align*} \left [\frac{{\left (B b + 2 \, A c\right )} \sqrt{c} x^{2} \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 (B c x^{2} - 2 \, A c\right )}}{4 \, c x^{2}}, -\frac{{\left (B b + 2 \, A c\right )} \sqrt{-c} x^{2} \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 (B c x^{2} - 2 \, A c\right )}}{2 \, c x^{2}}\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^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23202, size = 124, normalized size = 1.28 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2} + b} B x \mathrm{sgn}\left (x\right ) + \frac{2 \, A b \sqrt{c} \mathrm{sgn}\left (x\right )}{{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b} - \frac{{\left (B b \sqrt{c} \mathrm{sgn}\left (x\right ) + 2 \, A c^{\frac{3}{2}} \mathrm{sgn}\left (x\right )\right )} \log \left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2}\right )}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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