Optimal. Leaf size=25 \[ \frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}} \]
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Rubi [A] time = 0.0191952, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1247, 636} \[ \frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1247
Rule 636
Rubi steps
\begin{align*} \int \frac{x \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{2+3 x}{\left (3+5 x+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{8+11 x^2}{13 \sqrt{3+5 x^2+x^4}}\\ \end{align*}
Mathematica [A] time = 0.0954649, size = 25, normalized size = 1. \[ \frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 22, normalized size = 0.9 \begin{align*}{\frac{11\,{x}^{2}+8}{13}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952234, size = 43, normalized size = 1.72 \begin{align*} \frac{11 \, x^{2}}{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} + \frac{8}{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.33069, size = 113, normalized size = 4.52 \begin{align*} \frac{11 \, x^{4} + 55 \, x^{2} + \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (11 \, x^{2} + 8\right )} + 33}{13 \,{\left (x^{4} + 5 \, x^{2} + 3\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 \left (3 x^{2} + 2\right )}{\left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16028, size = 28, normalized size = 1.12 \begin{align*} \frac{11 \, x^{2} + 8}{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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