Integrand size = 24, antiderivative size = 198 \[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {742 x \left (2+x^2\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {x \left (36783+10643 x^2\right ) \sqrt {2+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}-\frac {742 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {13879 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{385 \sqrt {2+3 x^2+x^4}} \]
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Time = 0.06 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1220, 1190, 1203, 1113, 1149} \[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {13879 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{385 \sqrt {x^4+3 x^2+2}}-\frac {742 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{15 \sqrt {x^4+3 x^2+2}}+\frac {25}{11} x \left (x^4+3 x^2+2\right )^{5/2}+\frac {1}{693} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac {x \left (10643 x^2+36783\right ) \sqrt {x^4+3 x^2+2}}{1155}+\frac {742 x \left (x^2+2\right )}{15 \sqrt {x^4+3 x^2+2}} \]
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Rule 1113
Rule 1149
Rule 1190
Rule 1203
Rule 1220
Rubi steps \begin{align*} \text {integral}& = \frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {1}{11} \int \left (489+320 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx \\ & = \frac {1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {1}{231} \int \left (15684+10643 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx \\ & = \frac {x \left (36783+10643 x^2\right ) \sqrt {2+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {\int \frac {249822+171402 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{3465} \\ & = \frac {x \left (36783+10643 x^2\right ) \sqrt {2+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {742}{15} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {27758}{385} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {742 x \left (2+x^2\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {x \left (36783+10643 x^2\right ) \sqrt {2+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}-\frac {742 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {13879 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{385 \sqrt {2+3 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.63 \[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {429318 x+1160065 x^3+1333551 x^5+892084 x^7+363480 x^9+82075 x^{11}+7875 x^{13}-171402 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-78420 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{3465 \sqrt {2+3 x^2+x^4}} \]
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Result contains complex when optimal does not.
Time = 1.80 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {x \left (7875 x^{8}+58450 x^{6}+172380 x^{4}+258044 x^{2}+214659\right ) \sqrt {x^{4}+3 x^{2}+2}}{3465}-\frac {13879 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{385 \sqrt {x^{4}+3 x^{2}+2}}+\frac {371 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}\) | \(143\) |
default | \(\frac {11492 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{231}+\frac {258044 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{3465}+\frac {23851 x \sqrt {x^{4}+3 x^{2}+2}}{385}-\frac {13879 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{385 \sqrt {x^{4}+3 x^{2}+2}}+\frac {371 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}+\frac {25 x^{9} \sqrt {x^{4}+3 x^{2}+2}}{11}+\frac {1670 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{99}\) | \(189\) |
elliptic | \(\frac {11492 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{231}+\frac {258044 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{3465}+\frac {23851 x \sqrt {x^{4}+3 x^{2}+2}}{385}-\frac {13879 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{385 \sqrt {x^{4}+3 x^{2}+2}}+\frac {371 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}+\frac {25 x^{9} \sqrt {x^{4}+3 x^{2}+2}}{11}+\frac {1670 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{99}\) | \(189\) |
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Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.34 \[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {-171402 i \, x E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 421224 i \, x F(\arcsin \left (\frac {i}{x}\right )\,|\,2) + {\left (7875 \, x^{10} + 58450 \, x^{8} + 172380 \, x^{6} + 258044 \, x^{4} + 214659 \, x^{2} + 171402\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}}{3465 \, x} \]
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\[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{2}\, dx \]
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\[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2} \,d x } \]
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\[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2} \,d x } \]
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Timed out. \[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int {\left (5\,x^2+7\right )}^2\,{\left (x^4+3\,x^2+2\right )}^{3/2} \,d x \]
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