Optimal. Leaf size=168 \[ \frac {\left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \left (3 d^2-2 e^2\right ) F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {2 e x \left (x^2+2\right ) (d-e)}{\sqrt {x^4+3 x^2+2}}-\frac {2 \sqrt {2} e \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} (d-e) E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {x^4+3 x^2+2}}+\frac {1}{3} e^2 x \sqrt {x^4+3 x^2+2} \]
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Rubi [A] time = 0.07, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1206, 1189, 1099, 1135} \[ \frac {\left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \left (3 d^2-2 e^2\right ) F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {2 e x \left (x^2+2\right ) (d-e)}{\sqrt {x^4+3 x^2+2}}-\frac {2 \sqrt {2} e \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} (d-e) E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {x^4+3 x^2+2}}+\frac {1}{3} e^2 x \sqrt {x^4+3 x^2+2} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1189
Rule 1206
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2}{\sqrt {2+3 x^2+x^4}} \, dx &=\frac {1}{3} e^2 x \sqrt {2+3 x^2+x^4}+\frac {1}{3} \int \frac {3 d^2-2 e^2+6 (d-e) e x^2}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {1}{3} e^2 x \sqrt {2+3 x^2+x^4}+(2 (d-e) e) \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {1}{3} \left (3 d^2-2 e^2\right ) \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {2 (d-e) e x \left (2+x^2\right )}{\sqrt {2+3 x^2+x^4}}+\frac {1}{3} e^2 x \sqrt {2+3 x^2+x^4}-\frac {2 \sqrt {2} (d-e) e \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 {\left (3 d^2-2 e^2\right ) \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {2+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 127, normalized size = 0.76 \[ \frac {-i \sqrt {x^2+1} \sqrt {x^2+2} \left (3 d^2-6 d e+4 e^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-6 i e \sqrt {x^2+1} \sqrt {x^2+2} (d-e) E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+e^2 x \left (x^4+3 x^2+2\right )}{3 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}{\sqrt {x^{4} + 3 \, x^{2} + 2}}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{2}}{\sqrt {x^{4} + 3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 235, normalized size = 1.40 \[ -\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, d^{2} \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}+\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+\EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right ) d e}{\sqrt {x^{4}+3 x^{2}+2}}+\left (\frac {\sqrt {x^{4}+3 x^{2}+2}\, x}{3}+\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{3 \sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+\EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{\sqrt {x^{4}+3 x^{2}+2}}\right ) e^{2} \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{2}}{\sqrt {x^{4} + 3 \, x^{2} + 2}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^2}{\sqrt {x^4+3\,x^2+2}} \,d x \]
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 \[ \int \frac {\left (d + e x^{2}\right )^{2}}{\sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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