Optimal. Leaf size=100 \[ -\frac{b (b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{3/2}}-\frac{\sqrt{b x^2+c x^4} (b B-4 A c)}{8 c}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2} \]
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Rubi [A] time = 0.19702, antiderivative size = 100, 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, 794, 664, 620, 206} \[ -\frac{b (b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{3/2}}-\frac{\sqrt{b x^2+c x^4} (b B-4 A c)}{8 c}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 794
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} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) \sqrt{b x+c x^2}}{x} \, dx,x,x^2\right )\\ &=\frac{B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2}+\frac{\left (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 )}{4 c}\\ &=-\frac{(b B-4 A c) \sqrt{b x^2+c x^4}}{8 c}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2}-\frac{(b (b B-4 A c)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{16 c}\\ &=-\frac{(b B-4 A c) \sqrt{b x^2+c x^4}}{8 c}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2}-\frac{(b (b B-4 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 )}{8 c}\\ &=-\frac{(b B-4 A c) \sqrt{b x^2+c x^4}}{8 c}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2}-\frac{b (b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.143771, size = 91, normalized size = 0.91 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{c} \left (4 A c+b B+2 B c x^2\right )-\frac{\sqrt{b} (b B-4 A c) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{x \sqrt{\frac{c x^2}{b}+1}}\right )}{8 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 124, normalized size = 1.2 \begin{align*}{\frac{1}{8\,x}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 2\,B\sqrt{c} \left ( c{x}^{2}+b \right ) ^{3/2}x+4\,A{c}^{3/2}\sqrt{c{x}^{2}+b}x-B\sqrt{c}\sqrt{c{x}^{2}+b}xb+4\,A\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) bc-B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{2} \right ){c}^{-{\frac{3}{2}}}{\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(-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 = 1.17224, size = 392, normalized size = 3.92 \begin{align*} \left [-\frac{{\left (B b^{2} - 4 \, A b c\right )} \sqrt{c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \,{\left (2 \, B c^{2} x^{2} + B b c + 4 \, A c^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{16 \, c^{2}}, \frac{{\left (B b^{2} - 4 \, A b c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) +{\left (2 \, B c^{2} x^{2} + B b c + 4 \, A c^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{8 \, c^{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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34825, size = 139, normalized size = 1.39 \begin{align*} \frac{1}{8} \,{\left (2 \, B x^{2} \mathrm{sgn}\left (x\right ) + \frac{B b c \mathrm{sgn}\left (x\right ) + 4 \, A c^{2} \mathrm{sgn}\left (x\right )}{c^{2}}\right )} \sqrt{c x^{2} + b} x + \frac{{\left (B b^{2} \mathrm{sgn}\left (x\right ) - 4 \, A b c \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right )}{8 \, c^{\frac{3}{2}}} - \frac{{\left (B b^{2} \log \left ({\left | b \right |}\right ) - 4 \, A b c \log \left ({\left | b \right |}\right )\right )} \mathrm{sgn}\left (x\right )}{16 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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