Integrand size = 24, antiderivative size = 111 \[ \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )^2} \, dx=-\frac {A}{5 b^2 x^5}-\frac {b B-2 A c}{3 b^3 x^3}+\frac {c (2 b B-3 A c)}{b^4 x}+\frac {c^2 (b B-A c) x}{2 b^4 \left (b+c x^2\right )}+\frac {c^{3/2} (5 b B-7 A c) \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{9/2}} \]
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Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1598, 467, 1816, 211} \[ \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )^2} \, dx=\frac {c^{3/2} (5 b B-7 A c) \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{9/2}}+\frac {c^2 x (b B-A c)}{2 b^4 \left (b+c x^2\right )}+\frac {c (2 b B-3 A c)}{b^4 x}-\frac {b B-2 A c}{3 b^3 x^3}-\frac {A}{5 b^2 x^5} \]
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Rule 211
Rule 467
Rule 1598
Rule 1816
Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x^2}{x^6 \left (b+c x^2\right )^2} \, dx \\ & = \frac {c^2 (b B-A c) x}{2 b^4 \left (b+c x^2\right )}-\frac {1}{2} c^2 \int \frac {-\frac {2 A}{b c^2}-\frac {2 (b B-A c) x^2}{b^2 c^2}+\frac {2 (b B-A c) x^4}{b^3 c}-\frac {(b B-A c) x^6}{b^4}}{x^6 \left (b+c x^2\right )} \, dx \\ & = \frac {c^2 (b B-A c) x}{2 b^4 \left (b+c x^2\right )}-\frac {1}{2} c^2 \int \left (-\frac {2 A}{b^2 c^2 x^6}-\frac {2 (b B-2 A c)}{b^3 c^2 x^4}+\frac {2 (2 b B-3 A c)}{b^4 c x^2}+\frac {-5 b B+7 A c}{b^4 \left (b+c x^2\right )}\right ) \, dx \\ & = -\frac {A}{5 b^2 x^5}-\frac {b B-2 A c}{3 b^3 x^3}+\frac {c (2 b B-3 A c)}{b^4 x}+\frac {c^2 (b B-A c) x}{2 b^4 \left (b+c x^2\right )}+\frac {\left (c^2 (5 b B-7 A c)\right ) \int \frac {1}{b+c x^2} \, dx}{2 b^4} \\ & = -\frac {A}{5 b^2 x^5}-\frac {b B-2 A c}{3 b^3 x^3}+\frac {c (2 b B-3 A c)}{b^4 x}+\frac {c^2 (b B-A c) x}{2 b^4 \left (b+c x^2\right )}+\frac {c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{9/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )^2} \, dx=-\frac {A}{5 b^2 x^5}+\frac {-b B+2 A c}{3 b^3 x^3}+\frac {c (2 b B-3 A c)}{b^4 x}+\frac {c^2 (b B-A c) x}{2 b^4 \left (b+c x^2\right )}+\frac {c^{3/2} (5 b B-7 A c) \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{9/2}} \]
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Time = 1.83 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {A}{5 b^{2} x^{5}}-\frac {-2 A c +B b}{3 x^{3} b^{3}}-\frac {c \left (3 A c -2 B b \right )}{b^{4} x}-\frac {c^{2} \left (\frac {\left (\frac {A c}{2}-\frac {B b}{2}\right ) x}{c \,x^{2}+b}+\frac {\left (7 A c -5 B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{b^{4}}\) | \(99\) |
risch | \(\frac {-\frac {c^{2} \left (7 A c -5 B b \right ) x^{6}}{2 b^{4}}-\frac {c \left (7 A c -5 B b \right ) x^{4}}{3 b^{3}}+\frac {\left (7 A c -5 B b \right ) x^{2}}{15 b^{2}}-\frac {A}{5 b}}{x^{5} \left (c \,x^{2}+b \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{9} \textit {\_Z}^{2}+49 A^{2} c^{5}-70 A B b \,c^{4}+25 B^{2} b^{2} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} b^{9}+98 A^{2} c^{5}-140 A B b \,c^{4}+50 B^{2} b^{2} c^{3}\right ) x +\left (7 A \,b^{5} c^{2}-5 B \,b^{6} c \right ) \textit {\_R} \right )\right )}{4}\) | \(179\) |
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Time = 0.27 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.77 \[ \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )^2} \, dx=\left [\frac {30 \, {\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 20 \, {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} - 12 \, A b^{3} - 4 \, {\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2} - 15 \, {\left ({\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{7} + {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{5}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} - 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right )}{60 \, {\left (b^{4} c x^{7} + b^{5} x^{5}\right )}}, \frac {15 \, {\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 10 \, {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} - 6 \, A b^{3} - 2 \, {\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2} + 15 \, {\left ({\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{7} + {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{5}\right )} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right )}{30 \, {\left (b^{4} c x^{7} + b^{5} x^{5}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (104) = 208\).
Time = 0.36 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.96 \[ \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )^2} \, dx=- \frac {\sqrt {- \frac {c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right ) \log {\left (- \frac {b^{5} \sqrt {- \frac {c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right )}{- 7 A c^{3} + 5 B b c^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right ) \log {\left (\frac {b^{5} \sqrt {- \frac {c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right )}{- 7 A c^{3} + 5 B b c^{2}} + x \right )}}{4} + \frac {- 6 A b^{3} + x^{6} \left (- 105 A c^{3} + 75 B b c^{2}\right ) + x^{4} \left (- 70 A b c^{2} + 50 B b^{2} c\right ) + x^{2} \cdot \left (14 A b^{2} c - 10 B b^{3}\right )}{30 b^{5} x^{5} + 30 b^{4} c x^{7}} \]
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Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )^2} \, dx=\frac {15 \, {\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 10 \, {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} - 6 \, A b^{3} - 2 \, {\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2}}{30 \, {\left (b^{4} c x^{7} + b^{5} x^{5}\right )}} + \frac {{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )^2} \, dx=\frac {{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b^{4}} + \frac {B b c^{2} x - A c^{3} x}{2 \, {\left (c x^{2} + b\right )} b^{4}} + \frac {30 \, B b c x^{4} - 45 \, A c^{2} x^{4} - 5 \, B b^{2} x^{2} + 10 \, A b c x^{2} - 3 \, A b^{2}}{15 \, b^{4} x^{5}} \]
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Time = 8.99 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )^2} \, dx=-\frac {\frac {A}{5\,b}-\frac {x^2\,\left (7\,A\,c-5\,B\,b\right )}{15\,b^2}+\frac {c^2\,x^6\,\left (7\,A\,c-5\,B\,b\right )}{2\,b^4}+\frac {c\,x^4\,\left (7\,A\,c-5\,B\,b\right )}{3\,b^3}}{c\,x^7+b\,x^5}-\frac {c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (7\,A\,c-5\,B\,b\right )}{2\,b^{9/2}} \]
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