Optimal. Leaf size=68 \[ \frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]
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Rubi [A] time = 0.0539312, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1114, 640, 621, 206} \[ \frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{a+b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{4 c}\\ &=\frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{2 c}\\ &=\frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0140484, size = 68, normalized size = 1. \[ \frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.172, size = 56, normalized size = 0.8 \begin{align*}{\frac{1}{2\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{b}{4}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){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 = 1.62224, size = 383, normalized size = 5.63 \begin{align*} \left [\frac{b \sqrt{c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 4 \, a c\right ) + 4 \, \sqrt{c x^{4} + b x^{2} + a} c}{8 \, c^{2}}, \frac{b \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2} + a} c}{4 \, 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{x^{3}}{\sqrt{a + b x^{2} + c x^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19704, size = 82, normalized size = 1.21 \begin{align*} \frac{b \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{3}{2}}} + \frac{\sqrt{c x^{4} + b x^{2} + a}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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