Optimal. Leaf size=49 \[ \frac{3}{2} \sqrt{x^4+5 x^2+3}-\frac{11}{4} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]
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Rubi [A] time = 0.0323246, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1247, 640, 621, 206} \[ \frac{3}{2} \sqrt{x^4+5 x^2+3}-\frac{11}{4} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 1247
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x \left (2+3 x^2\right )}{\sqrt{3+5 x^2+x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{2+3 x}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac{3}{2} \sqrt{3+5 x^2+x^4}-\frac{11}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac{3}{2} \sqrt{3+5 x^2+x^4}-\frac{11}{2} \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{5+2 x^2}{\sqrt{3+5 x^2+x^4}}\right )\\ &=\frac{3}{2} \sqrt{3+5 x^2+x^4}-\frac{11}{4} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.0095622, size = 49, normalized size = 1. \[ \frac{3}{2} \sqrt{x^4+5 x^2+3}-\frac{11}{4} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 36, normalized size = 0.7 \begin{align*}{\frac{3}{2}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{11}{4}\ln \left ({\frac{5}{2}}+{x}^{2}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952726, size = 53, normalized size = 1.08 \begin{align*} \frac{3}{2} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{11}{4} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39899, size = 103, normalized size = 2.1 \begin{align*} \frac{3}{2} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{11}{4} \, \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\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 )}{\sqrt{x^{4} + 5 x^{2} + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11169, size = 53, normalized size = 1.08 \begin{align*} \frac{3}{2} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{11}{4} \, \log \left (2 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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