Optimal. Leaf size=235 \[ \frac {625 \sqrt {x^4+3 x^2+2} x}{504 \left (5 x^2+7\right )}-\frac {31 \left (x^2+2\right ) x}{56 \sqrt {x^4+3 x^2+2}}+\frac {\left (11 x^2+20\right ) x}{36 \sqrt {x^4+3 x^2+2}}-\frac {463 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{336 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {31 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{28 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {375 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{784 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
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Rubi [A] time = 0.43, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1228, 1178, 1189, 1099, 1135, 1223, 1716, 1214, 1456, 539} \[ \frac {625 \sqrt {x^4+3 x^2+2} x}{504 \left (5 x^2+7\right )}-\frac {31 \left (x^2+2\right ) x}{56 \sqrt {x^4+3 x^2+2}}+\frac {\left (11 x^2+20\right ) x}{36 \sqrt {x^4+3 x^2+2}}-\frac {463 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{336 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {31 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{28 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {375 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{784 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 539
Rule 1099
Rule 1135
Rule 1178
Rule 1189
Rule 1214
Rule 1223
Rule 1228
Rule 1456
Rule 1716
Rubi steps
\begin {align*} \int \frac {1}{\left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2}} \, dx &=\int \left (\frac {14+5 x^2}{36 \left (2+3 x^2+x^4\right )^{3/2}}-\frac {25}{6 \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}}-\frac {25}{36 \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}}\right ) \, dx\\ &=\frac {1}{36} \int \frac {14+5 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx-\frac {25}{36} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx-\frac {25}{6} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \left (20+11 x^2\right )}{36 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{504 \left (7+5 x^2\right )}-\frac {1}{72} \int \frac {26+22 x^2}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {25}{504} \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}{72} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {125}{144} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\\ &=\frac {x \left (20+11 x^2\right )}{36 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{504 \left (7+5 x^2\right )}-\frac {25 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{72 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {1}{504} \int \frac {-175-125 x^2}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {11}{36} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {13}{36} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {325}{504} \int \frac {1}{\left (7+5 x^2\right ) \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}{144 \sqrt {2+3 x^2+x^4}}\\ &=-\frac {11 x \left (2+x^2\right )}{36 \sqrt {2+3 x^2+x^4}}+\frac {x \left (20+11 x^2\right )}{36 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{504 \left (7+5 x^2\right )}+\frac {11 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{18 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {17 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{24 \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 )}{504 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}-\frac {125}{504} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {325 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{1008}-\frac {25}{72} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {1625 \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{2016}\\ &=-\frac {31 x \left (2+x^2\right )}{56 \sqrt {2+3 x^2+x^4}}+\frac {x \left (20+11 x^2\right )}{36 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{504 \left (7+5 x^2\right )}+\frac {31 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{28 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {463 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{336 \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 )}{504 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {\left (1625 \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}{2016 \sqrt {2+3 x^2+x^4}}\\ &=-\frac {31 x \left (2+x^2\right )}{56 \sqrt {2+3 x^2+x^4}}+\frac {x \left (20+11 x^2\right )}{36 \sqrt {2+3 x^2+x^4}}+\frac {625 x \sqrt {2+3 x^2+x^4}}{504 \left (7+5 x^2\right )}+\frac {31 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{28 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {463 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{336 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {375 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{784 \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 = 0.28, size = 208, normalized size = 0.89 \[ \frac {3255 x^5+10157 x^3+182 i \sqrt {x^2+1} \sqrt {x^2+2} \left (5 x^2+7\right ) F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+651 i \sqrt {x^2+1} \sqrt {x^2+2} \left (5 x^2+7\right ) E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+1125 i \sqrt {x^2+1} \sqrt {x^2+2} x^2 \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+1575 i \sqrt {x^2+1} \sqrt {x^2+2} \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+7490 x}{1176 \left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 3 \, x^{2} + 2}}{25 \, x^{12} + 220 \, x^{10} + 794 \, x^{8} + 1504 \, x^{6} + 1577 \, x^{4} + 868 \, x^{2} + 196}, x\right ) \]
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 \[ \int \frac {1}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 185, normalized size = 0.79 \[ \frac {625 \sqrt {x^{4}+3 x^{2}+2}\, x}{504 \left (5 x^{2}+7\right )}+\frac {31 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{112 \sqrt {x^{4}+3 x^{2}+2}}+\frac {13 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{168 \sqrt {x^{4}+3 x^{2}+2}}+\frac {75 i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{392 \sqrt {x^{4}+3 x^{2}+2}}-\frac {2 \left (-\frac {11}{72} x^{3}-\frac {5}{18} x \right )}{\sqrt {x^{4}+3 x^{2}+2}} \]
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 \[ \int \frac {1}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\left (5\,x^2+7\right )}^2\,{\left (x^4+3\,x^2+2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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