Integrand size = 15, antiderivative size = 68 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2} x^3}+\frac {1}{b^2 \sqrt {a+\frac {b}{x^2}} x}-\frac {\text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{b^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {342, 294, 223, 212} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{b^{5/2}}+\frac {1}{b^2 x \sqrt {a+\frac {b}{x^2}}}+\frac {1}{3 b x^3 \left (a+\frac {b}{x^2}\right )^{3/2}} \]
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Rule 212
Rule 223
Rule 294
Rule 342
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right )^{5/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2} x^3}-\frac {\text {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{b} \\ & = \frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2} x^3}+\frac {1}{b^2 \sqrt {a+\frac {b}{x^2}} x}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right )}{b^2} \\ & = \frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2} x^3}+\frac {1}{b^2 \sqrt {a+\frac {b}{x^2}} x}-\frac {\text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^2}} x}\right )}{b^2} \\ & = \frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2} x^3}+\frac {1}{b^2 \sqrt {a+\frac {b}{x^2}} x}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\frac {\sqrt {b} \left (4 b+3 a x^2\right )-3 \left (b+a x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{3 b^{5/2} \sqrt {a+\frac {b}{x^2}} x \left (b+a x^2\right )} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {\left (a \,x^{2}+b \right ) \left (3 x^{2} a \,b^{\frac {3}{2}}+4 b^{\frac {5}{2}}-3 \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) \left (a \,x^{2}+b \right )^{\frac {3}{2}} b \right )}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} x^{5} b^{\frac {7}{2}}}\) | \(77\) |
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Time = 0.34 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.32 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\left [\frac {3 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt {b} \log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \, {\left (3 \, a b x^{3} + 4 \, b^{2} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{6 \, {\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}}, \frac {3 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (3 \, a b x^{3} + 4 \, b^{2} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (58) = 116\).
Time = 1.78 (sec) , antiderivative size = 740, normalized size of antiderivative = 10.88 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\frac {3 a^{3} b^{4} x^{6} \log {\left (\frac {a x^{2}}{b} \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} - \frac {6 a^{3} b^{4} x^{6} \log {\left (\sqrt {\frac {a x^{2}}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {6 a^{2} b^{5} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {9 a^{2} b^{5} x^{4} \log {\left (\frac {a x^{2}}{b} \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} - \frac {18 a^{2} b^{5} x^{4} \log {\left (\sqrt {\frac {a x^{2}}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {14 a b^{6} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {9 a b^{6} x^{2} \log {\left (\frac {a x^{2}}{b} \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} - \frac {18 a b^{6} x^{2} \log {\left (\sqrt {\frac {a x^{2}}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {8 b^{7} \sqrt {\frac {a x^{2}}{b} + 1}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} + \frac {3 b^{7} \log {\left (\frac {a x^{2}}{b} \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} - \frac {6 b^{7} \log {\left (\sqrt {\frac {a x^{2}}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {13}{2}} x^{6} + 18 a^{2} b^{\frac {15}{2}} x^{4} + 18 a b^{\frac {17}{2}} x^{2} + 6 b^{\frac {19}{2}}} \]
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Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {3 \, {\left (a + \frac {b}{x^{2}}\right )} x^{2} + b}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b^{2} x^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=-\frac {{\left (3 \, \sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b}\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {-b} b^{\frac {5}{2}}} + \frac {\arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2} \mathrm {sgn}\left (x\right )} + \frac {3 \, a x^{2} + 4 \, b}{3 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} b^{2} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx=\int \frac {1}{x^6\,{\left (a+\frac {b}{x^2}\right )}^{5/2}} \,d x \]
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