Integrand size = 25, antiderivative size = 305 \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx=\frac {32 x \left (5+\sqrt {13}+2 x^2\right )}{9 \sqrt {3+5 x^2+x^4}}-\frac {64 \sqrt {3+5 x^2+x^4}}{9 x}-\frac {\left (2-9 x^2\right ) \sqrt {3+5 x^2+x^4}}{3 x^3}-\frac {16 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{9 \sqrt {3+5 x^2+x^4}}+\frac {49 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{3 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]
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Time = 0.08 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1285, 1295, 1203, 1113, 1149} \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx=\frac {49 \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{3 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {x^4+5 x^2+3}}-\frac {16 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{9 \sqrt {x^4+5 x^2+3}}-\frac {64 \sqrt {x^4+5 x^2+3}}{9 x}+\frac {32 x \left (2 x^2+\sqrt {13}+5\right )}{9 \sqrt {x^4+5 x^2+3}}-\frac {\sqrt {x^4+5 x^2+3} \left (2-9 x^2\right )}{3 x^3} \]
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Rule 1113
Rule 1149
Rule 1203
Rule 1285
Rule 1295
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (2-9 x^2\right ) \sqrt {3+5 x^2+x^4}}{3 x^3}-\frac {1}{3} \int \frac {-64-49 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {64 \sqrt {3+5 x^2+x^4}}{9 x}-\frac {\left (2-9 x^2\right ) \sqrt {3+5 x^2+x^4}}{3 x^3}+\frac {1}{9} \int \frac {147+64 x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {64 \sqrt {3+5 x^2+x^4}}{9 x}-\frac {\left (2-9 x^2\right ) \sqrt {3+5 x^2+x^4}}{3 x^3}+\frac {64}{9} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx+\frac {49}{3} \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {32 x \left (5+\sqrt {13}+2 x^2\right )}{9 \sqrt {3+5 x^2+x^4}}-\frac {64 \sqrt {3+5 x^2+x^4}}{9 x}-\frac {\left (2-9 x^2\right ) \sqrt {3+5 x^2+x^4}}{3 x^3}-\frac {16 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{9 \sqrt {3+5 x^2+x^4}}+\frac {49 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{3 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 9.68 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.78 \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx=\frac {-2 \left (18+141 x^2+191 x^4+37 x^6\right )+32 i \sqrt {2} \left (-5+\sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-13+32 \sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{18 x^3 \sqrt {3+5 x^2+x^4}} \]
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Time = 2.15 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {37 \sqrt {x^{4}+5 x^{2}+3}}{9 x}+\frac {98 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {256 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{3 x^{3}}\) | \(228\) |
risch | \(-\frac {37 x^{6}+191 x^{4}+141 x^{2}+18}{9 x^{3} \sqrt {x^{4}+5 x^{2}+3}}+\frac {98 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {256 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(228\) |
elliptic | \(-\frac {37 \sqrt {x^{4}+5 x^{2}+3}}{9 x}+\frac {98 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {256 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{3 x^{3}}\) | \(228\) |
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\[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx=\int { \frac {\sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx=\int \frac {\left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}}{x^{4}}\, dx \]
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\[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx=\int { \frac {\sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx=\int { \frac {\sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx=\int \frac {\left (3\,x^2+2\right )\,\sqrt {x^4+5\,x^2+3}}{x^4} \,d x \]
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