Optimal. Leaf size=148 \[ \frac{\sqrt{3 x^2+2} \text{EllipticF}\left (\tan ^{-1}(2 x),\frac{5}{8}\right )}{2 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}}+\frac{4 \sqrt{3 x^2+2} x}{3 \sqrt{4 x^2+1}}-\frac{2 \sqrt{2} \sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]
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Rubi [A] time = 0.0497495, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {422, 418, 492, 411} \[ \frac{4 \sqrt{3 x^2+2} x}{3 \sqrt{4 x^2+1}}+\frac{\sqrt{3 x^2+2} F\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{2 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}}-\frac{2 \sqrt{2} \sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 422
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\sqrt{1+4 x^2}}{\sqrt{2+3 x^2}} \, dx &=4 \int \frac{x^2}{\sqrt{2+3 x^2} \sqrt{1+4 x^2}} \, dx+\int \frac{1}{\sqrt{2+3 x^2} \sqrt{1+4 x^2}} \, dx\\ &=\frac{4 x \sqrt{2+3 x^2}}{3 \sqrt{1+4 x^2}}+\frac{\sqrt{2+3 x^2} F\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{2 \sqrt{2} \sqrt{\frac{2+3 x^2}{1+4 x^2}} \sqrt{1+4 x^2}}-\frac{4}{3} \int \frac{\sqrt{2+3 x^2}}{\left (1+4 x^2\right )^{3/2}} \, dx\\ &=\frac{4 x \sqrt{2+3 x^2}}{3 \sqrt{1+4 x^2}}-\frac{2 \sqrt{2} \sqrt{2+3 x^2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{\frac{2+3 x^2}{1+4 x^2}} \sqrt{1+4 x^2}}+\frac{\sqrt{2+3 x^2} F\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{2 \sqrt{2} \sqrt{\frac{2+3 x^2}{1+4 x^2}} \sqrt{1+4 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0045624, size = 27, normalized size = 0.18 \[ -\frac{i E\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{8}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 20, normalized size = 0.1 \begin{align*} -{\frac{i}{3}}{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{6},{\frac{2\,\sqrt{6}}{3}} \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{\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\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{\sqrt{4 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{\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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