Optimal. Leaf size=316 \[ \frac {e^2 x \sqrt {x^4+3 x^2+2}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}-\frac {e x \left (x^2+2\right )}{2 d \sqrt {x^4+3 x^2+2} \left (d^2-3 d e+2 e^2\right )}-\frac {e \left (x^2+2\right ) \left (3 d^2-6 d e+2 e^2\right ) \Pi \left (1-\frac {e}{d};\tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {2} d^2 \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2} (d-2 e) (d-e)^2}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^2+2}{2 x^2+2}} (2 d-e) F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{2 d \sqrt {x^4+3 x^2+2} (d-e)^2}+\frac {e \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} d \sqrt {x^4+3 x^2+2} (d-2 e) (d-e)} \]
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Rubi [A] time = 0.33, antiderivative size = 399, normalized size of antiderivative = 1.26, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1223, 1716, 1189, 1099, 1135, 1214, 1456, 539} \[ \frac {e^2 x \sqrt {x^4+3 x^2+2}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}-\frac {e x \left (x^2+2\right )}{2 d \sqrt {x^4+3 x^2+2} \left (d^2-3 d e+2 e^2\right )}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \left (3 d^2-6 d e+2 e^2\right ) F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {2} d \sqrt {x^4+3 x^2+2} (d-2 e) (d-e)^2}-\frac {e \left (x^2+2\right ) \left (3 d^2-6 d e+2 e^2\right ) \Pi \left (1-\frac {e}{d};\tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {2} d^2 \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2} (d-2 e) (d-e)^2}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4+3 x^2+2} (d-2 e) (d-e)}+\frac {e \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} d \sqrt {x^4+3 x^2+2} (d-2 e) (d-e)} \]
Antiderivative was successfully verified.
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Rule 539
Rule 1099
Rule 1135
Rule 1189
Rule 1214
Rule 1223
Rule 1456
Rule 1716
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx &=\frac {e^2 x \sqrt {2+3 x^2+x^4}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}-\frac {\int \frac {-2 \left (d^2-3 d e+e^2\right )+2 d e x^2+e^2 x^4}{\left (d+e x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{2 d (d-2 e) (d-e)}\\ &=\frac {e^2 x \sqrt {2+3 x^2+x^4}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}+\frac {\int \frac {-d e^2-e^3 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{2 d (d-2 e) (d-e) e^2}+\frac {\left (3 d^2-6 d e+2 e^2\right ) \int \frac {1}{\left (d+e x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{2 d (d-2 e) (d-e)}\\ &=\frac {e^2 x \sqrt {2+3 x^2+x^4}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}-\frac {\int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{2 (d-2 e) (d-e)}+\frac {\left (3 d^2-6 d e+2 e^2\right ) \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{2 d (d-2 e) (d-e)^2}-\frac {\left (e \left (3 d^2-6 d e+2 e^2\right )\right ) \int \frac {2+2 x^2}{\left (d+e x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{4 d (d-2 e) (d-e)^2}-\frac {e \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{2 d \left (d^2-3 d e+2 e^2\right )}\\ &=-\frac {e x \left (2+x^2\right )}{2 d \left (d^2-3 d e+2 e^2\right ) \sqrt {2+3 x^2+x^4}}+\frac {e^2 x \sqrt {2+3 x^2+x^4}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}+\frac {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} d (d-2 e) (d-e) \sqrt {2+3 x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {2} (d-2 e) (d-e) \sqrt {2+3 x^2+x^4}}+\frac {\left (3 d^2-6 d e+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 )}{2 \sqrt {2} d (d-2 e) (d-e)^2 \sqrt {2+3 x^2+x^4}}-\frac {\left (e \left (3 d^2-6 d e+2 e^2\right ) \sqrt {1+\frac {x^2}{2}} \sqrt {2+2 x^2}\right ) \int \frac {\sqrt {2+2 x^2}}{\sqrt {1+\frac {x^2}{2}} \left (d+e x^2\right )} \, dx}{4 d (d-2 e) (d-e)^2 \sqrt {2+3 x^2+x^4}}\\ &=-\frac {e x \left (2+x^2\right )}{2 d \left (d^2-3 d e+2 e^2\right ) \sqrt {2+3 x^2+x^4}}+\frac {e^2 x \sqrt {2+3 x^2+x^4}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}+\frac {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} d (d-2 e) (d-e) \sqrt {2+3 x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {2} (d-2 e) (d-e) \sqrt {2+3 x^2+x^4}}+\frac {\left (3 d^2-6 d e+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 )}{2 \sqrt {2} d (d-2 e) (d-e)^2 \sqrt {2+3 x^2+x^4}}-\frac {e \left (3 d^2-6 d e+2 e^2\right ) \left (2+x^2\right ) \Pi \left (1-\frac {e}{d};\tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {2} d^2 (d-2 e) (d-e)^2 \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.60, size = 175, normalized size = 0.55 \[ \frac {\frac {e^2 x \left (x^4+3 x^2+2\right )}{\left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}+\frac {i \sqrt {x^2+1} \sqrt {x^2+2} \left (\left (-3 d^2+6 d e-2 e^2\right ) \Pi \left (\frac {2 e}{d};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+d (d-e) F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+d e E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )\right )}{d (d-2 e) (d-e)}}{2 d \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{e^{2} x^{8} + {\left (2 \, d e + 3 \, e^{2}\right )} x^{6} + {\left (d^{2} + 6 \, d e + 2 \, e^{2}\right )} x^{4} + {\left (3 \, d^{2} + 4 \, d e\right )} x^{2} + 2 \, d^{2}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{4} + 3 \, x^{2} + 2} {\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 443, normalized size = 1.40 \[ \frac {\sqrt {x^{4}+3 x^{2}+2}\, e^{2} x}{2 \left (d^{2}-3 d e +2 e^{2}\right ) \left (e \,x^{2}+d \right ) d}+\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, e \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{4 \left (d^{2}-3 d e +2 e^{2}\right ) \sqrt {x^{4}+3 x^{2}+2}\, d}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, e \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{4 \left (d^{2}-3 d e +2 e^{2}\right ) \sqrt {x^{4}+3 x^{2}+2}\, d}+\frac {3 i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, e \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 e}{d}, \sqrt {2}\right )}{\left (d^{2}-3 d e +2 e^{2}\right ) \sqrt {x^{4}+3 x^{2}+2}\, d}-\frac {i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, e^{2} \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 e}{d}, \sqrt {2}\right )}{\left (d^{2}-3 d e +2 e^{2}\right ) \sqrt {x^{4}+3 x^{2}+2}\, d^{2}}+\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{4 \left (d^{2}-3 d e +2 e^{2}\right ) \sqrt {x^{4}+3 x^{2}+2}}-\frac {3 i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 e}{d}, \sqrt {2}\right )}{2 \left (d^{2}-3 d e +2 e^{2}\right ) \sqrt {x^{4}+3 x^{2}+2}} \]
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 \[ \int \frac {1}{\sqrt {x^{4} + 3 \, x^{2} + 2} {\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (e\,x^2+d\right )}^2\,\sqrt {x^4+3\,x^2+2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (d + e x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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