Integrand size = 24, antiderivative size = 222 \[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\frac {9 x \left (2+x^2\right )}{175 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {9 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{175 \sqrt {2+3 x^2+x^4}}+\frac {59 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{2+2 x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{1050 \sqrt {2+3 x^2+x^4}}+\frac {9 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{2+2 x^2}} \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{2450 \sqrt {2+3 x^2+x^4}} \]
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Time = 0.29 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.50, number of steps used = 21, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1242, 1113, 1149, 1136, 1203, 1237, 1730, 1228, 1470, 553} \[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\frac {44 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{1875 \sqrt {x^4+3 x^2+2}}+\frac {81 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{8750 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {9 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{175 \sqrt {x^4+3 x^2+2}}+\frac {3 \sqrt {2} \left (x^2+2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{875 \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}}-\frac {39 \left (x^2+2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{12250 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}}-\frac {3 \sqrt {x^4+3 x^2+2} x}{175 \left (5 x^2+7\right )}+\frac {1}{75} \sqrt {x^4+3 x^2+2} x+\frac {9 \left (x^2+2\right ) x}{175 \sqrt {x^4+3 x^2+2}} \]
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Rule 553
Rule 1113
Rule 1136
Rule 1149
Rule 1203
Rule 1228
Rule 1237
Rule 1242
Rule 1470
Rule 1730
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {52}{625 \sqrt {2+3 x^2+x^4}}+\frac {16 x^2}{125 \sqrt {2+3 x^2+x^4}}+\frac {x^4}{25 \sqrt {2+3 x^2+x^4}}+\frac {36}{625 \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}}-\frac {12}{625 \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}}\right ) \, dx \\ & = -\left (\frac {12}{625} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx\right )+\frac {1}{25} \int \frac {x^4}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {36}{625} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx+\frac {52}{625} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {16}{125} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {16 x \left (2+x^2\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {16 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {26 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{625 \sqrt {2+3 x^2+x^4}}+\frac {3 \int \frac {62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{4375}-\frac {6}{625} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {1}{75} \int \frac {2+6 x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {3}{125} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {16 x \left (2+x^2\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {16 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {23 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{625 \sqrt {2+3 x^2+x^4}}-\frac {3 \int \frac {-175-125 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{109375}+\frac {39 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{4375}-\frac {2}{75} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {2}{25} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\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}{125 \sqrt {2+3 x^2+x^4}} \\ & = \frac {6 x \left (2+x^2\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {6 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {44 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1875 \sqrt {2+3 x^2+x^4}}+\frac {3 \sqrt {2} \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {3}{875} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {39 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{8750}+\frac {3}{625} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {39 \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{3500} \\ & = \frac {9 x \left (2+x^2\right )}{175 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {9 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{175 \sqrt {2+3 x^2+x^4}}+\frac {81 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{8750 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {44 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1875 \sqrt {2+3 x^2+x^4}}+\frac {3 \sqrt {2} \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}-\frac {\left (39 \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}{3500 \sqrt {2+3 x^2+x^4}} \\ & = \frac {9 x \left (2+x^2\right )}{175 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {9 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{175 \sqrt {2+3 x^2+x^4}}+\frac {81 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{8750 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {44 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1875 \sqrt {2+3 x^2+x^4}}-\frac {39 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{12250 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {3 \sqrt {2} \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.28 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.96 \[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\frac {2800 x+6650 x^3+5075 x^5+1225 x^7-945 i \sqrt {1+x^2} \sqrt {2+x^2} \left (7+5 x^2\right ) E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-182 i \sqrt {1+x^2} \sqrt {2+x^2} \left (7+5 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )+189 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticPi}\left (\frac {10}{7},i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )+135 i x^2 \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticPi}\left (\frac {10}{7},i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{18375 \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \]
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Result contains complex when optimal does not.
Time = 2.64 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {3 x \sqrt {x^{4}+3 x^{2}+2}}{175 \left (5 x^{2}+7\right )}+\frac {x \sqrt {x^{4}+3 x^{2}+2}}{75}-\frac {13 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2625 \sqrt {x^{4}+3 x^{2}+2}}-\frac {9 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{350 \sqrt {x^{4}+3 x^{2}+2}}+\frac {9 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{6125 \sqrt {x^{4}+3 x^{2}+2}}\) | \(177\) |
elliptic | \(-\frac {3 x \sqrt {x^{4}+3 x^{2}+2}}{175 \left (5 x^{2}+7\right )}+\frac {x \sqrt {x^{4}+3 x^{2}+2}}{75}-\frac {13 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2625 \sqrt {x^{4}+3 x^{2}+2}}-\frac {9 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{350 \sqrt {x^{4}+3 x^{2}+2}}+\frac {9 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{6125 \sqrt {x^{4}+3 x^{2}+2}}\) | \(177\) |
risch | \(\frac {\sqrt {x^{4}+3 x^{2}+2}\, x \left (7 x^{2}+8\right )}{525 x^{2}+735}-\frac {23 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{750 \sqrt {x^{4}+3 x^{2}+2}}+\frac {9 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{350 \sqrt {x^{4}+3 x^{2}+2}}+\frac {9 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{6125 \sqrt {x^{4}+3 x^{2}+2}}\) | \(183\) |
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\[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int \frac {\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}}}{\left (5 x^{2} + 7\right )^{2}}\, dx \]
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\[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int \frac {{\left (x^4+3\,x^2+2\right )}^{3/2}}{{\left (5\,x^2+7\right )}^2} \,d x \]
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