Optimal. Leaf size=80 \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+1}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{3 \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}} \]
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Rubi [A] time = 0.0275457, antiderivative size = 80, 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} \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+1}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{3 \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
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Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{1+x^2} \sqrt{2+3 x^2}} \, dx &=\frac{x \sqrt{2+3 x^2}}{3 \sqrt{1+x^2}}-\frac{1}{3} \int \frac{\sqrt{2+3 x^2}}{\left (1+x^2\right )^{3/2}} \, dx\\ &=\frac{x \sqrt{2+3 x^2}}{3 \sqrt{1+x^2}}-\frac{\sqrt{2} \sqrt{2+3 x^2} E\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{3 \sqrt{1+x^2} \sqrt{\frac{2+3 x^2}{1+x^2}}}\\ \end{align*}
Mathematica [C] time = 0.0279188, size = 34, normalized size = 0.42 \[ -\frac{1}{3} i \sqrt{2} \left (E\left (i \sinh ^{-1}(x)|\frac{3}{2}\right )-\text{EllipticF}\left (i \sinh ^{-1}(x),\frac{3}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 30, normalized size = 0.4 \begin{align*}{\frac{i}{3}} \left ({\it EllipticF} \left ( ix,{\frac{\sqrt{6}}{2}} \right ) -{\it EllipticE} \left ( ix,{\frac{\sqrt{6}}{2}} \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} + 1}}\,{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} + 1} x^{2}}{3 \, x^{4} + 5 \, 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{x^{2} + 1} \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} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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