Optimal. Leaf size=173 \[ \frac {x}{6 \sqrt {x^4+3 x^2+2}}-\frac {9 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{2 x^2+2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {x^4+3 x^2+2}}+\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {x^4+3 x^2+2}}+\frac {125 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{84 \sqrt {2} \sqrt {x^4+3 x^2+2}} \]
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Rubi [A] time = 0.14, antiderivative size = 207, normalized size of antiderivative = 1.20, 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 = {1221, 1178, 1189, 1099, 1135, 1214, 1456, 539} \[ -\frac {x \left (x^2+2\right )}{3 \sqrt {x^4+3 x^2+2}}+\frac {x \left (2 x^2+5\right )}{6 \sqrt {x^4+3 x^2+2}}-\frac {9 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {x^4+3 x^2+2}}+\frac {125 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{84 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 539
Rule 1099
Rule 1135
Rule 1178
Rule 1189
Rule 1214
Rule 1221
Rule 1456
Rubi steps
\begin {align*} \int \frac {1}{\left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}} \, dx &=-\left (\frac {1}{6} \int \frac {-8-5 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx\right )-\frac {25}{6} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \left (5+2 x^2\right )}{6 \sqrt {2+3 x^2+x^4}}+\frac {1}{12} \int \frac {-2-4 x^2}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {25}{12} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {125}{24} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \left (5+2 x^2\right )}{6 \sqrt {2+3 x^2+x^4}}-\frac {25 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {1}{6} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {1}{3} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {\left (125 \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}{24 \sqrt {2+3 x^2+x^4}}\\ &=-\frac {x \left (2+x^2\right )}{3 \sqrt {2+3 x^2+x^4}}+\frac {x \left (5+2 x^2\right )}{6 \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 )}{3 \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 )}{4 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {125 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{84 \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 = 0.17, size = 138, normalized size = 0.80 \[ \frac {14 x^3-7 i \sqrt {x^2+1} \sqrt {x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+14 i \sqrt {x^2+1} \sqrt {x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+25 i \sqrt {x^2+1} \sqrt {x^2+2} \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+35 x}{42 \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{5 \, x^{10} + 37 \, x^{8} + 107 \, x^{6} + 151 \, x^{4} + 104 \, x^{2} + 28}, x\right ) \]
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 \[ \int \frac {1}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 161, normalized size = 0.93 \[ \frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{12 \sqrt {x^{4}+3 x^{2}+2}}+\frac {25 i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{42 \sqrt {x^{4}+3 x^{2}+2}}-\frac {2 \left (-\frac {1}{6} x^{3}-\frac {5}{12} x \right )}{\sqrt {x^{4}+3 x^{2}+2}} \]
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 \[ \int \frac {1}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (5\,x^2+7\right )\,{\left (x^4+3\,x^2+2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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