Optimal. Leaf size=121 \[ \frac {5 x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}+\frac {7 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {5 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {x^4+3 x^2+2}} \]
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Rubi [A] time = 0.03, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1189, 1099, 1135} \[ \frac {5 x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}+\frac {7 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {5 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1189
Rubi steps
\begin {align*} \int \frac {7+5 x^2}{\sqrt {2+3 x^2+x^4}} \, dx &=5 \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+7 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {5 x \left (2+x^2\right )}{\sqrt {2+3 x^2+x^4}}-\frac {5 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2+3 x^2+x^4}}+\frac {7 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 69, normalized size = 0.57 \[ -\frac {i \sqrt {x^2+1} \sqrt {x^2+2} \left (2 F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+5 E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )\right )}{\sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {5 \, x^{2} + 7}{\sqrt {x^{4} + 3 \, x^{2} + 2}}, x\right ) \]
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 \[ \int \frac {5 \, x^{2} + 7}{\sqrt {x^{4} + 3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 106, normalized size = 0.88 \[ -\frac {7 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}+\frac {5 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+\EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}} \]
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 \[ \int \frac {5 \, x^{2} + 7}{\sqrt {x^{4} + 3 \, x^{2} + 2}}\,{d x} \]
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 \[ \int \frac {5\,x^2+7}{\sqrt {x^4+3\,x^2+2}} \,d x \]
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 \[ \int \frac {5 x^{2} + 7}{\sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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