Optimal. Leaf size=36 \[ \frac{2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]
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Rubi [A] time = 0.028675, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1114, 636} \[ \frac{2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 636
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.0955075, size = 36, normalized size = 1. \[ \frac{2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 38, normalized size = 1.1 \begin{align*} -{\frac{b{x}^{2}+2\,a}{4\,ac-{b}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \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 = 1.71841, size = 139, normalized size = 3.86 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}} \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}}{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28001, size = 59, normalized size = 1.64 \begin{align*} \frac{\frac{b x^{2}}{b^{2} - 4 \, a c} + \frac{2 \, a}{b^{2} - 4 \, a c}}{\sqrt{c x^{4} + b x^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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