Optimal. Leaf size=67 \[ \frac {\sqrt {2+x^2} \sqrt {-1+3 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {7}{2}} x}{\sqrt {-1+3 x^2}}\right )|\frac {6}{7}\right )}{\sqrt {7} \sqrt {-2+5 x^2+3 x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1111}
\begin {gather*} \frac {\sqrt {x^2+2} \sqrt {3 x^2-1} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {7}{2}} x}{\sqrt {3 x^2-1}}\right )|\frac {6}{7}\right )}{\sqrt {7} \sqrt {3 x^4+5 x^2-2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1111
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2+5 x^2+3 x^4}} \, dx &=\frac {\sqrt {2+x^2} \sqrt {-1+3 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {7}{2}} x}{\sqrt {-1+3 x^2}}\right )|\frac {6}{7}\right )}{\sqrt {7} \sqrt {-2+5 x^2+3 x^4}}\\ \end {align*}
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Mathematica [A]
time = 10.03, size = 54, normalized size = 0.81 \begin {gather*} \frac {\sqrt {1-3 x^2} \sqrt {2+x^2} F\left (\sin ^{-1}\left (\sqrt {3} x\right )|-\frac {1}{6}\right )}{\sqrt {6} \sqrt {-2+5 x^2+3 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.02, size = 53, normalized size = 0.79
method | result | size |
default | \(-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, i \sqrt {6}\right )}{2 \sqrt {3 x^{4}+5 x^{2}-2}}\) | \(53\) |
elliptic | \(-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, i \sqrt {6}\right )}{2 \sqrt {3 x^{4}+5 x^{2}-2}}\) | \(53\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {3 x^{4} + 5 x^{2} - 2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {3\,x^4+5\,x^2-2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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