Optimal. Leaf size=51 \[ \frac {\sqrt {5 x^2+2} \operatorname {EllipticF}\left (\tan ^{-1}(x),-\frac {3}{2}\right )}{\sqrt {2} \sqrt {x^2+1} \sqrt {\frac {5 x^2+2}{x^2+1}}} \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {418} \[ \frac {\sqrt {5 x^2+2} F\left (\tan ^{-1}(x)|-\frac {3}{2}\right )}{\sqrt {2} \sqrt {x^2+1} \sqrt {\frac {5 x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
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Rule 418
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x^2} \sqrt {2+5 x^2}} \, dx &=\frac {\sqrt {2+5 x^2} F\left (\tan ^{-1}(x)|-\frac {3}{2}\right )}{\sqrt {2} \sqrt {1+x^2} \sqrt {\frac {2+5 x^2}{1+x^2}}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 19, normalized size = 0.37 \[ -\frac {i \operatorname {EllipticF}\left (i \sinh ^{-1}(x),\frac {5}{2}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x^{2} + 2} \sqrt {x^{2} + 1}}{5 \, x^{4} + 7 \, x^{2} + 2}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {5 \, x^{2} + 2} \sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 17, normalized size = 0.33 \[ -\frac {i \sqrt {2}\, \EllipticF \left (i x , \frac {\sqrt {10}}{2}\right )}{2} \]
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 \[ \int \frac {1}{\sqrt {5 \, x^{2} + 2} \sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {x^2+1}\,\sqrt {5\,x^2+2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {x^{2} + 1} \sqrt {5 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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