Optimal. Leaf size=148 \[ \frac {\sqrt {3 x^2+2} \operatorname {EllipticF}\left (\tan ^{-1}(2 x),\frac {5}{8}\right )}{2 \sqrt {2} \sqrt {\frac {3 x^2+2}{4 x^2+1}} \sqrt {4 x^2+1}}+\frac {4 \sqrt {3 x^2+2} x}{3 \sqrt {4 x^2+1}}-\frac {2 \sqrt {2} \sqrt {3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac {5}{8}\right )}{3 \sqrt {\frac {3 x^2+2}{4 x^2+1}} \sqrt {4 x^2+1}} \]
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Rubi [A] time = 0.05, antiderivative size = 148, normalized size of antiderivative = 1.00, 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 411
Rule 418
Rule 422
Rule 492
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*}
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Mathematica [C] time = 0.00, 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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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {4 \, x^{2} + 1}}{\sqrt {3 \, x^{2} + 2}}, x\right ) \]
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 \[ \int \frac {\sqrt {4 \, x^{2} + 1}}{\sqrt {3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 20, normalized size = 0.14 \[ -\frac {i \sqrt {3}\, \EllipticE \left (\frac {i \sqrt {6}\, x}{2}, \frac {2 \sqrt {6}}{3}\right )}{3} \]
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 \[ \int \frac {\sqrt {4 \, x^{2} + 1}}{\sqrt {3 \, x^{2} + 2}}\,{d x} \]
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 \[ \int \frac {\sqrt {4\,x^2+1}}{\sqrt {3\,x^2+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {4 x^{2} + 1}}{\sqrt {3 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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