Optimal. Leaf size=68 \[ \frac{\left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{8 c}-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{3/2}} \]
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Rubi [A] time = 0.0638366, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2013, 612, 620, 206} \[ \frac{\left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{8 c}-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2013
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int x \sqrt{b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{\left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{8 c}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{16 c}\\ &=\frac{\left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{8 c}-\frac{b^2 \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{\left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{8 c}-\frac{b^2 \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 = 0.0501411, size = 90, normalized size = 1.32 \[ \frac{x \sqrt{b+c x^2} \left (\sqrt{c} x \sqrt{b+c x^2} \left (b+2 c x^2\right )-b^2 \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )\right )}{8 c^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 84, normalized size = 1.2 \begin{align*}{\frac{1}{8\,x}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 2\,x \left ( c{x}^{2}+b \right ) ^{3/2}\sqrt{c}-\sqrt{c}\sqrt{c{x}^{2}+b}xb-\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}{c}^{-{\frac{3}{2}}}} \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 = 1.61851, size = 315, normalized size = 4.63 \begin{align*} \left [\frac{b^{2} \sqrt{c} \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 (2 \, c^{2} x^{2} + b c\right )}}{16 \, c^{2}}, \frac{b^{2} \sqrt{-c} \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 (2 \, c^{2} x^{2} + b c\right )}}{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 x \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 [A] time = 1.25786, size = 93, normalized size = 1.37 \begin{align*} \frac{1}{8} \, \sqrt{c x^{2} + b}{\left (2 \, x^{2} \mathrm{sgn}\left (x\right ) + \frac{b \mathrm{sgn}\left (x\right )}{c}\right )} x + \frac{b^{2} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right ) \mathrm{sgn}\left (x\right )}{8 \, c^{\frac{3}{2}}} - \frac{b^{2} \log \left ({\left | b \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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