Optimal. Leaf size=82 \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+4}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left (\left .\tan ^{-1}\left (\frac{x}{2}\right )\right |-5\right )}{3 \sqrt{x^2+4} \sqrt{\frac{3 x^2+2}{x^2+4}}} \]
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Rubi [A] time = 0.0291869, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {492, 411} \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+4}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left (\left .\tan ^{-1}\left (\frac{x}{2}\right )\right |-5\right )}{3 \sqrt{x^2+4} \sqrt{\frac{3 x^2+2}{x^2+4}}} \]
Antiderivative was successfully verified.
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Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{4+x^2} \sqrt{2+3 x^2}} \, dx &=\frac{x \sqrt{2+3 x^2}}{3 \sqrt{4+x^2}}-\frac{4}{3} \int \frac{\sqrt{2+3 x^2}}{\left (4+x^2\right )^{3/2}} \, dx\\ &=\frac{x \sqrt{2+3 x^2}}{3 \sqrt{4+x^2}}-\frac{\sqrt{2} \sqrt{2+3 x^2} E\left (\left .\tan ^{-1}\left (\frac{x}{2}\right )\right |-5\right )}{3 \sqrt{4+x^2} \sqrt{\frac{2+3 x^2}{4+x^2}}}\\ \end{align*}
Mathematica [C] time = 0.0280067, size = 38, normalized size = 0.46 \[ -\frac{1}{3} i \sqrt{2} \left (E\left (\left .i \sinh ^{-1}\left (\frac{x}{2}\right )\right |6\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{2}\right ),6\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 26, normalized size = 0.3 \begin{align*}{\frac{i}{3}} \left ({\it EllipticF} \left ({\frac{i}{2}}x,\sqrt{6} \right ) -{\it EllipticE} \left ({\frac{i}{2}}x,\sqrt{6} \right ) \right ) \sqrt{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 4} x^{2}}{3 \, x^{4} + 14 \, x^{2} + 8}, x\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^{2}}{\sqrt{x^{2} + 4} \sqrt{3 x^{2} + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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