Optimal. Leaf size=209 \[ -\frac {25 \sqrt {x^4+3 x^2+2} x}{84 \left (5 x^2+7\right )}+\frac {5 \left (x^2+2\right ) x}{84 \sqrt {x^4+3 x^2+2}}+\frac {9 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{56 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{42 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {65 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{1176 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
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Rubi [A] time = 0.19, antiderivative size = 209, normalized size of antiderivative = 1.00, 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} \[ -\frac {25 \sqrt {x^4+3 x^2+2} x}{84 \left (5 x^2+7\right )}+\frac {5 \left (x^2+2\right ) x}{84 \sqrt {x^4+3 x^2+2}}+\frac {9 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{56 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{42 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {65 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{1176 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 539
Rule 1099
Rule 1135
Rule 1189
Rule 1214
Rule 1223
Rule 1456
Rule 1716
Rubi steps
\begin {align*} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx &=-\frac {25 x \sqrt {2+3 x^2+x^4}}{84 \left (7+5 x^2\right )}+\frac {1}{84} \int \frac {62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\\ &=-\frac {25 x \sqrt {2+3 x^2+x^4}}{84 \left (7+5 x^2\right )}-\frac {\int \frac {-175-125 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{2100}+\frac {13}{84} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\\ &=-\frac {25 x \sqrt {2+3 x^2+x^4}}{84 \left (7+5 x^2\right )}+\frac {5}{84} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {13}{168} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {1}{12} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {65}{336} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {5 x \left (2+x^2\right )}{84 \sqrt {2+3 x^2+x^4}}-\frac {25 x \sqrt {2+3 x^2+x^4}}{84 \left (7+5 x^2\right )}-\frac {5 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{42 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {9 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{56 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {\left (65 \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}{336 \sqrt {2+3 x^2+x^4}}\\ &=\frac {5 x \left (2+x^2\right )}{84 \sqrt {2+3 x^2+x^4}}-\frac {25 x \sqrt {2+3 x^2+x^4}}{84 \left (7+5 x^2\right )}-\frac {5 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{42 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {9 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{56 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {65 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{1176 \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 = 0.26, size = 208, normalized size = 1.00 \[ \frac {-175 x^5-525 x^3-14 i \sqrt {x^2+1} \sqrt {x^2+2} \left (5 x^2+7\right ) F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-35 i \sqrt {x^2+1} \sqrt {x^2+2} \left (5 x^2+7\right ) E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-65 i \sqrt {x^2+1} \sqrt {x^2+2} x^2 \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-91 i \sqrt {x^2+1} \sqrt {x^2+2} \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-350 x}{588 \left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{25 \, x^{8} + 145 \, x^{6} + 309 \, x^{4} + 287 \, x^{2} + 98}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 162, normalized size = 0.78 \[ -\frac {25 \sqrt {x^{4}+3 x^{2}+2}\, x}{84 \left (5 x^{2}+7\right )}-\frac {5 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{168 \sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{84 \sqrt {x^{4}+3 x^{2}+2}}-\frac {13 i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{588 \sqrt {x^{4}+3 x^{2}+2}} \]
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 \[ \int \frac {1}{\sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\left (5\,x^2+7\right )}^2\,\sqrt {x^4+3\,x^2+2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (5 x^{2} + 7\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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