Optimal. Leaf size=70 \[ -\frac{b \tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{4 c^{3/2}}-\frac{\sqrt{a+b x^2-c x^4}}{2 c} \]
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Rubi [A] time = 0.0581514, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {1114, 640, 621, 204} \[ -\frac{b \tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{4 c^{3/2}}-\frac{\sqrt{a+b x^2-c x^4}}{2 c} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 640
Rule 621
Rule 204
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 \tan ^{-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 = 0.0160885, size = 70, normalized size = 1. \[ -\frac{b \tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{4 c^{3/2}}-\frac{\sqrt{a+b x^2-c x^4}}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.161, size = 58, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,c}\sqrt{-c{x}^{4}+b{x}^{2}+a}}+{\frac{b}{4}\arctan \left ({\sqrt{c} \left ({x}^{2}-{\frac{b}{2\,c}} \right ){\frac{1}{\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 = 1.56805, size = 390, normalized size = 5.57 \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 = 1.24795, size = 95, normalized size = 1.36 \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 \, \sqrt{-c} c} - \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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