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 = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 411
Rule 492
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*}
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Mathematica [C] time = 0.04, size = 50, normalized size = 0.57 \[ -\frac {i \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {8}{3}\right )-F\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {8}{3}\right )\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {4 \, x^{2} + 1} \sqrt {3 \, x^{2} + 2} x^{2}}{12 \, x^{4} + 11 \, x^{2} + 2}, x\right ) \]
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 \[ \int \frac {x^{2}}{\sqrt {4 \, x^{2} + 1} \sqrt {3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 36, normalized size = 0.41 \[ \frac {i \left (-\EllipticE \left (\frac {i \sqrt {6}\, x}{2}, \frac {2 \sqrt {6}}{3}\right )+\EllipticF \left (\frac {i \sqrt {6}\, x}{2}, \frac {2 \sqrt {6}}{3}\right )\right ) \sqrt {3}}{12} \]
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 \[ \int \frac {x^{2}}{\sqrt {4 \, x^{2} + 1} \sqrt {3 \, x^{2} + 2}}\,{d x} \]
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 \[ \int \frac {x^2}{\sqrt {3\,x^2+2}\,\sqrt {4\,x^2+1}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {3 x^{2} + 2} \sqrt {4 x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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