Optimal. Leaf size=88 \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{4 x^2+1}}-\frac{\sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]
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Rubi [A] time = 0.0307442, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {492, 411} \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{4 x^2+1}}-\frac{\sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{2+3 x^2} \sqrt{1+4 x^2}} \, dx &=\frac{x \sqrt{2+3 x^2}}{3 \sqrt{1+4 x^2}}-\frac{1}{3} \int \frac{\sqrt{2+3 x^2}}{\left (1+4 x^2\right )^{3/2}} \, dx\\ &=\frac{x \sqrt{2+3 x^2}}{3 \sqrt{1+4 x^2}}-\frac{\sqrt{2+3 x^2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{2} \sqrt{\frac{2+3 x^2}{1+4 x^2}} \sqrt{1+4 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0325587, size = 50, normalized size = 0.57 \[ -\frac{i \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{8}{3}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),\frac{8}{3}\right )\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 36, normalized size = 0.4 \begin{align*}{\frac{i}{12}} \left ({\it EllipticF} \left ({\frac{i}{2}}x\sqrt{6},{\frac{2\,\sqrt{6}}{3}} \right ) -{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{6},{\frac{2\,\sqrt{6}}{3}} \right ) \right ) \sqrt{3} \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{4 \, x^{2} + 1} \sqrt{3 \, x^{2} + 2}}\,{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{4 \, x^{2} + 1} \sqrt{3 \, x^{2} + 2} x^{2}}{12 \, x^{4} + 11 \, x^{2} + 2}, 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{3 x^{2} + 2} \sqrt{4 x^{2} + 1}}\, 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{4 \, x^{2} + 1} \sqrt{3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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