Integrand size = 25, antiderivative size = 331 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^6} \, dx=\frac {361 x \left (5+\sqrt {13}+2 x^2\right )}{15 \sqrt {3+5 x^2+x^4}}-\frac {722 \sqrt {3+5 x^2+x^4}}{15 x}-\frac {\left (40-87 x^2\right ) \sqrt {3+5 x^2+x^4}}{5 x^3}-\frac {\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}-\frac {361 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{15 \sqrt {3+5 x^2+x^4}}+\frac {103 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]
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Time = 0.11 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1285, 1295, 1203, 1113, 1149} \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^6} \, dx=\frac {103 \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {x^4+5 x^2+3}}-\frac {361 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{15 \sqrt {x^4+5 x^2+3}}-\frac {722 \sqrt {x^4+5 x^2+3}}{15 x}+\frac {361 x \left (2 x^2+\sqrt {13}+5\right )}{15 \sqrt {x^4+5 x^2+3}}-\frac {\left (2-5 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{5 x^5}-\frac {\left (40-87 x^2\right ) \sqrt {x^4+5 x^2+3}}{5 x^3} \]
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Rule 1113
Rule 1149
Rule 1203
Rule 1285
Rule 1295
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}-\frac {1}{5} \int \frac {\left (-120-87 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx \\ & = -\frac {\left (40-87 x^2\right ) \sqrt {3+5 x^2+x^4}}{5 x^3}-\frac {\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}+\frac {1}{15} \int \frac {2166+1545 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {722 \sqrt {3+5 x^2+x^4}}{15 x}-\frac {\left (40-87 x^2\right ) \sqrt {3+5 x^2+x^4}}{5 x^3}-\frac {\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}-\frac {1}{45} \int \frac {-4635-2166 x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {722 \sqrt {3+5 x^2+x^4}}{15 x}-\frac {\left (40-87 x^2\right ) \sqrt {3+5 x^2+x^4}}{5 x^3}-\frac {\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}+\frac {722}{15} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx+103 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {361 x \left (5+\sqrt {13}+2 x^2\right )}{15 \sqrt {3+5 x^2+x^4}}-\frac {722 \sqrt {3+5 x^2+x^4}}{15 x}-\frac {\left (40-87 x^2\right ) \sqrt {3+5 x^2+x^4}}{5 x^3}-\frac {\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}-\frac {361 \sqrt {\frac {1}{6} \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 )}{15 \sqrt {3+5 x^2+x^4}}+\frac {103 \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 )}{\sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.27 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.74 \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^6} \, dx=\frac {-108-810 x^2-3438 x^4-4040 x^6-634 x^8+30 x^{10}+361 i \sqrt {2} \left (-5+\sqrt {13}\right ) x^5 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-260+361 \sqrt {13}\right ) x^5 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{30 x^5 \sqrt {3+5 x^2+x^4}} \]
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Time = 4.09 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {15 x^{10}-317 x^{8}-2020 x^{6}-1719 x^{4}-405 x^{2}-54}{15 x^{5} \sqrt {x^{4}+5 x^{2}+3}}+\frac {618 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {8664 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(238\) |
default | \(-\frac {6 \sqrt {x^{4}+5 x^{2}+3}}{5 x^{5}}-\frac {7 \sqrt {x^{4}+5 x^{2}+3}}{x^{3}}-\frac {392 \sqrt {x^{4}+5 x^{2}+3}}{15 x}+\frac {618 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {8664 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}+x \sqrt {x^{4}+5 x^{2}+3}\) | \(259\) |
elliptic | \(-\frac {6 \sqrt {x^{4}+5 x^{2}+3}}{5 x^{5}}-\frac {7 \sqrt {x^{4}+5 x^{2}+3}}{x^{3}}-\frac {392 \sqrt {x^{4}+5 x^{2}+3}}{15 x}+\frac {618 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {8664 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}+x \sqrt {x^{4}+5 x^{2}+3}\) | \(259\) |
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\[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )}}{x^{6}} \,d x } \]
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\[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^6} \, dx=\int \frac {\left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
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\[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )}}{x^{6}} \,d x } \]
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\[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )}}{x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^6} \, dx=\int \frac {\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{x^6} \,d x \]
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