Integrand size = 20, antiderivative size = 117 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=-\frac {A}{3 a^3 x^3}+\frac {3 A b-a B}{a^4 x}+\frac {b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}+\frac {b (11 A b-7 a B) x}{8 a^4 \left (a+b x^2\right )}+\frac {5 \sqrt {b} (7 A b-3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2}} \]
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Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {467, 1273, 1275, 211} \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {5 \sqrt {b} (7 A b-3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2}}+\frac {b x (11 A b-7 a B)}{8 a^4 \left (a+b x^2\right )}+\frac {3 A b-a B}{a^4 x}+\frac {b x (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}-\frac {A}{3 a^3 x^3} \]
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Rule 211
Rule 467
Rule 1273
Rule 1275
Rubi steps \begin{align*} \text {integral}& = \frac {b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}-\frac {1}{4} b \int \frac {-\frac {4 A}{a b}+\frac {4 (A b-a B) x^2}{a^2 b}-\frac {3 (A b-a B) x^4}{a^3}}{x^4 \left (a+b x^2\right )^2} \, dx \\ & = \frac {b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}+\frac {b (11 A b-7 a B) x}{8 a^4 \left (a+b x^2\right )}-\frac {\int \frac {-8 a A b+8 b (2 A b-a B) x^2-\frac {b^2 (11 A b-7 a B) x^4}{a}}{x^4 \left (a+b x^2\right )} \, dx}{8 a^3 b} \\ & = \frac {b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}+\frac {b (11 A b-7 a B) x}{8 a^4 \left (a+b x^2\right )}-\frac {\int \left (-\frac {8 A b}{x^4}-\frac {8 b (-3 A b+a B)}{a x^2}+\frac {5 b^2 (-7 A b+3 a B)}{a \left (a+b x^2\right )}\right ) \, dx}{8 a^3 b} \\ & = -\frac {A}{3 a^3 x^3}+\frac {3 A b-a B}{a^4 x}+\frac {b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}+\frac {b (11 A b-7 a B) x}{8 a^4 \left (a+b x^2\right )}+\frac {(5 b (7 A b-3 a B)) \int \frac {1}{a+b x^2} \, dx}{8 a^4} \\ & = -\frac {A}{3 a^3 x^3}+\frac {3 A b-a B}{a^4 x}+\frac {b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}+\frac {b (11 A b-7 a B) x}{8 a^4 \left (a+b x^2\right )}+\frac {5 \sqrt {b} (7 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {105 A b^3 x^6+a^2 b x^2 \left (56 A-75 B x^2\right )+5 a b^2 x^4 \left (35 A-9 B x^2\right )-8 a^3 \left (A+3 B x^2\right )}{24 a^4 x^3 \left (a+b x^2\right )^2}+\frac {5 \sqrt {b} (7 A b-3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2}} \]
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Time = 2.54 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {A}{3 a^{3} x^{3}}-\frac {-3 A b +B a}{a^{4} x}+\frac {b \left (\frac {\left (\frac {11}{8} b^{2} A -\frac {7}{8} a b B \right ) x^{3}+\frac {a \left (13 A b -9 B a \right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (7 A b -3 B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}\) | \(98\) |
risch | \(\frac {\frac {5 b^{2} \left (7 A b -3 B a \right ) x^{6}}{8 a^{4}}+\frac {25 b \left (7 A b -3 B a \right ) x^{4}}{24 a^{3}}+\frac {\left (7 A b -3 B a \right ) x^{2}}{3 a^{2}}-\frac {A}{3 a}}{x^{3} \left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} \textit {\_Z}^{2}+49 A^{2} b^{3}-42 A B a \,b^{2}+9 B^{2} a^{2} b \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{9}+98 A^{2} b^{3}-84 A B a \,b^{2}+18 B^{2} a^{2} b \right ) x +\left (-7 A \,a^{5} b +3 B \,a^{6}\right ) \textit {\_R} \right )\right )}{16}\) | \(172\) |
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Time = 0.29 (sec) , antiderivative size = 368, normalized size of antiderivative = 3.15 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=\left [-\frac {30 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 50 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 16 \, A a^{3} + 16 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{48 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}, -\frac {15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{24 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (109) = 218\).
Time = 0.42 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.93 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {5 \sqrt {- \frac {b}{a^{9}}} \left (- 7 A b + 3 B a\right ) \log {\left (- \frac {5 a^{5} \sqrt {- \frac {b}{a^{9}}} \left (- 7 A b + 3 B a\right )}{- 35 A b^{2} + 15 B a b} + x \right )}}{16} - \frac {5 \sqrt {- \frac {b}{a^{9}}} \left (- 7 A b + 3 B a\right ) \log {\left (\frac {5 a^{5} \sqrt {- \frac {b}{a^{9}}} \left (- 7 A b + 3 B a\right )}{- 35 A b^{2} + 15 B a b} + x \right )}}{16} + \frac {- 8 A a^{3} + x^{6} \cdot \left (105 A b^{3} - 45 B a b^{2}\right ) + x^{4} \cdot \left (175 A a b^{2} - 75 B a^{2} b\right ) + x^{2} \cdot \left (56 A a^{2} b - 24 B a^{3}\right )}{24 a^{6} x^{3} + 48 a^{5} b x^{5} + 24 a^{4} b^{2} x^{7}} \]
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Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=-\frac {15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}}{24 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}} - \frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=-\frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} - \frac {7 \, B a b^{2} x^{3} - 11 \, A b^{3} x^{3} + 9 \, B a^{2} b x - 13 \, A a b^{2} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{4}} - \frac {3 \, B a x^{2} - 9 \, A b x^{2} + A a}{3 \, a^{4} x^{3}} \]
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Time = 5.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {\frac {x^2\,\left (7\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {A}{3\,a}+\frac {5\,b^2\,x^6\,\left (7\,A\,b-3\,B\,a\right )}{8\,a^4}+\frac {25\,b\,x^4\,\left (7\,A\,b-3\,B\,a\right )}{24\,a^3}}{a^2\,x^3+2\,a\,b\,x^5+b^2\,x^7}+\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (7\,A\,b-3\,B\,a\right )}{8\,a^{9/2}} \]
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