Integrand size = 24, antiderivative size = 117 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {A}{3 b^3 x^3}-\frac {b B-3 A c}{b^4 x}-\frac {c (b B-A c) x}{4 b^3 \left (b+c x^2\right )^2}-\frac {c (7 b B-11 A c) x}{8 b^4 \left (b+c x^2\right )}-\frac {5 \sqrt {c} (3 b B-7 A c) \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{9/2}} \]
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Time = 0.13 (sec) , antiderivative size = 117, 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 = {1598, 467, 1273, 1275, 211} \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {5 \sqrt {c} (3 b B-7 A c) \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{9/2}}-\frac {c x (7 b B-11 A c)}{8 b^4 \left (b+c x^2\right )}-\frac {b B-3 A c}{b^4 x}-\frac {c x (b B-A c)}{4 b^3 \left (b+c x^2\right )^2}-\frac {A}{3 b^3 x^3} \]
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Rule 211
Rule 467
Rule 1273
Rule 1275
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x^2}{x^4 \left (b+c x^2\right )^3} \, dx \\ & = -\frac {c (b B-A c) x}{4 b^3 \left (b+c x^2\right )^2}-\frac {1}{4} c \int \frac {-\frac {4 A}{b c}-\frac {4 (b B-A c) x^2}{b^2 c}+\frac {3 (b B-A c) x^4}{b^3}}{x^4 \left (b+c x^2\right )^2} \, dx \\ & = -\frac {c (b B-A c) x}{4 b^3 \left (b+c x^2\right )^2}-\frac {c (7 b B-11 A c) x}{8 b^4 \left (b+c x^2\right )}-\frac {\int \frac {-8 A b c-8 c (b B-2 A c) x^2+\frac {c^2 (7 b B-11 A c) x^4}{b}}{x^4 \left (b+c x^2\right )} \, dx}{8 b^3 c} \\ & = -\frac {c (b B-A c) x}{4 b^3 \left (b+c x^2\right )^2}-\frac {c (7 b B-11 A c) x}{8 b^4 \left (b+c x^2\right )}-\frac {\int \left (-\frac {8 A c}{x^4}-\frac {8 c (b B-3 A c)}{b x^2}+\frac {5 c^2 (3 b B-7 A c)}{b \left (b+c x^2\right )}\right ) \, dx}{8 b^3 c} \\ & = -\frac {A}{3 b^3 x^3}-\frac {b B-3 A c}{b^4 x}-\frac {c (b B-A c) x}{4 b^3 \left (b+c x^2\right )^2}-\frac {c (7 b B-11 A c) x}{8 b^4 \left (b+c x^2\right )}-\frac {(5 c (3 b B-7 A c)) \int \frac {1}{b+c x^2} \, dx}{8 b^4} \\ & = -\frac {A}{3 b^3 x^3}-\frac {b B-3 A c}{b^4 x}-\frac {c (b B-A c) x}{4 b^3 \left (b+c x^2\right )^2}-\frac {c (7 b B-11 A c) x}{8 b^4 \left (b+c x^2\right )}-\frac {5 \sqrt {c} (3 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{9/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {A}{3 b^3 x^3}+\frac {-b B+3 A c}{b^4 x}-\frac {c (b B-A c) x}{4 b^3 \left (b+c x^2\right )^2}-\frac {\left (7 b B c-11 A c^2\right ) x}{8 b^4 \left (b+c x^2\right )}-\frac {5 \sqrt {c} (3 b B-7 A c) \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{9/2}} \]
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Time = 1.79 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {A}{3 b^{3} x^{3}}-\frac {-3 A c +B b}{b^{4} x}+\frac {c \left (\frac {\left (\frac {11}{8} A \,c^{2}-\frac {7}{8} B b c \right ) x^{3}+\frac {b \left (13 A c -9 B b \right ) x}{8}}{\left (c \,x^{2}+b \right )^{2}}+\frac {5 \left (7 A c -3 B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}}\right )}{b^{4}}\) | \(98\) |
risch | \(\frac {\frac {5 c^{2} \left (7 A c -3 B b \right ) x^{6}}{8 b^{4}}+\frac {25 c \left (7 A c -3 B b \right ) x^{4}}{24 b^{3}}+\frac {\left (7 A c -3 B b \right ) x^{2}}{3 b^{2}}-\frac {A}{3 b}}{x^{3} \left (c \,x^{2}+b \right )^{2}}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{9} \textit {\_Z}^{2}+49 A^{2} c^{3}-42 A B b \,c^{2}+9 B^{2} b^{2} c \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} b^{9}+98 A^{2} c^{3}-84 A B b \,c^{2}+18 B^{2} b^{2} c \right ) x +\left (-7 A \,b^{5} c +3 B \,b^{6}\right ) \textit {\_R} \right )\right )}{16}\) | \(172\) |
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Time = 0.27 (sec) , antiderivative size = 368, normalized size of antiderivative = 3.15 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\left [-\frac {30 \, {\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 50 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} + 16 \, A b^{3} + 16 \, {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2} + 15 \, {\left ({\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{7} + 2 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{5} + {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{3}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} + 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right )}{48 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}}, -\frac {15 \, {\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 25 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} + 8 \, A b^{3} + 8 \, {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2} + 15 \, {\left ({\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{7} + 2 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{5} + {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{3}\right )} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right )}{24 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (109) = 218\).
Time = 0.40 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.93 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {5 \sqrt {- \frac {c}{b^{9}}} \left (- 7 A c + 3 B b\right ) \log {\left (- \frac {5 b^{5} \sqrt {- \frac {c}{b^{9}}} \left (- 7 A c + 3 B b\right )}{- 35 A c^{2} + 15 B b c} + x \right )}}{16} - \frac {5 \sqrt {- \frac {c}{b^{9}}} \left (- 7 A c + 3 B b\right ) \log {\left (\frac {5 b^{5} \sqrt {- \frac {c}{b^{9}}} \left (- 7 A c + 3 B b\right )}{- 35 A c^{2} + 15 B b c} + x \right )}}{16} + \frac {- 8 A b^{3} + x^{6} \cdot \left (105 A c^{3} - 45 B b c^{2}\right ) + x^{4} \cdot \left (175 A b c^{2} - 75 B b^{2} c\right ) + x^{2} \cdot \left (56 A b^{2} c - 24 B b^{3}\right )}{24 b^{6} x^{3} + 48 b^{5} c x^{5} + 24 b^{4} c^{2} x^{7}} \]
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Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {15 \, {\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 25 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} + 8 \, A b^{3} + 8 \, {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2}}{24 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}} - \frac {5 \, {\left (3 \, B b c - 7 \, A c^{2}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {5 \, {\left (3 \, B b c - 7 \, A c^{2}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{4}} - \frac {7 \, B b c^{2} x^{3} - 11 \, A c^{3} x^{3} + 9 \, B b^{2} c x - 13 \, A b c^{2} x}{8 \, {\left (c x^{2} + b\right )}^{2} b^{4}} - \frac {3 \, B b x^{2} - 9 \, A c x^{2} + A b}{3 \, b^{4} x^{3}} \]
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Time = 9.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {x^2\,\left (7\,A\,c-3\,B\,b\right )}{3\,b^2}-\frac {A}{3\,b}+\frac {5\,c^2\,x^6\,\left (7\,A\,c-3\,B\,b\right )}{8\,b^4}+\frac {25\,c\,x^4\,\left (7\,A\,c-3\,B\,b\right )}{24\,b^3}}{b^2\,x^3+2\,b\,c\,x^5+c^2\,x^7}+\frac {5\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (7\,A\,c-3\,B\,b\right )}{8\,b^{9/2}} \]
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