Optimal. Leaf size=178 \[ \frac {x \left (x^2+2\right )}{5 \sqrt {x^4+3 x^2+2}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^2+2}{2 x^2+2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {x^4+3 x^2+2}}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{35 \sqrt {2} \sqrt {x^4+3 x^2+2}} \]
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Rubi [A] time = 0.12, antiderivative size = 232, normalized size of antiderivative = 1.30, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1208, 1189, 1099, 1135, 1214, 1456, 539} \[ \frac {x \left (x^2+2\right )}{5 \sqrt {x^4+3 x^2+2}}+\frac {4 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{25 \sqrt {x^4+3 x^2+2}}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{25 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {x^4+3 x^2+2}}+\frac {3 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{35 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 539
Rule 1099
Rule 1135
Rule 1189
Rule 1208
Rule 1214
Rule 1456
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x^2+x^4}}{7+5 x^2} \, dx &=-\left (\frac {1}{25} \int \frac {-8-5 x^2}{\sqrt {2+3 x^2+x^4}} \, dx\right )-\frac {6}{25} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\\ &=-\left (\frac {3}{25} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx\right )+\frac {1}{5} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {3}{10} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx+\frac {8}{25} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \left (2+x^2\right )}{5 \sqrt {2+3 x^2+x^4}}-\frac {\sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {2+3 x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{25 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {4 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{25 \sqrt {2+3 x^2+x^4}}+\frac {\left (3 \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 (7+5 x^2\right )} \, dx}{10 \sqrt {2+3 x^2+x^4}}\\ &=\frac {x \left (2+x^2\right )}{5 \sqrt {2+3 x^2+x^4}}-\frac {\sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {2+3 x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{25 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {4 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{25 \sqrt {2+3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{35 \sqrt {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 = 0.15, size = 90, normalized size = 0.51 \[ -\frac {i \sqrt {x^2+1} \sqrt {x^2+2} \left (21 F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+35 E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-6 \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )\right )}{175 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{5 \, x^{2} + 7}, x\right ) \]
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 \[ \int \frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{5 \, x^{2} + 7}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 138, normalized size = 0.78 \[ -\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{10 \sqrt {x^{4}+3 x^{2}+2}}-\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{50 \sqrt {x^{4}+3 x^{2}+2}}+\frac {6 i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{175 \sqrt {x^{4}+3 x^{2}+2}} \]
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 \[ \int \frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{5 \, x^{2} + 7}\,{d x} \]
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 \[ \int \frac {\sqrt {x^4+3\,x^2+2}}{5\,x^2+7} \,d x \]
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 \[ \int \frac {\sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )}}{5 x^{2} + 7}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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