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 = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 411
Rule 492
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*}
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Mathematica [C] time = 0.03, size = 38, normalized size = 0.46 \[ -\frac {1}{3} i \sqrt {2} \left (E\left (\left .i \sinh ^{-1}\left (\frac {x}{2}\right )\right |6\right )-F\left (\left .i \sinh ^{-1}\left (\frac {x}{2}\right )\right |6\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {3 \, x^{2} + 2} \sqrt {x^{2} + 4} x^{2}}{3 \, x^{4} + 14 \, x^{2} + 8}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {x^{2} + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 26, normalized size = 0.32 \[ \frac {i \left (-\EllipticE \left (\frac {i x}{2}, \sqrt {6}\right )+\EllipticF \left (\frac {i x}{2}, \sqrt {6}\right )\right ) \sqrt {2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {x^{2} + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{\sqrt {x^2+4}\,\sqrt {3\,x^2+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {x^{2} + 4} \sqrt {3 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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