Integrand size = 20, antiderivative size = 90 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {A}{3 a^2 x^3}+\frac {2 A b-a B}{a^3 x}+\frac {b (A b-a B) x}{2 a^3 \left (a+b x^2\right )}+\frac {\sqrt {b} (5 A b-3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}} \]
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Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {467, 1275, 211} \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {b} (5 A b-3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}}+\frac {b x (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac {2 A b-a B}{a^3 x}-\frac {A}{3 a^2 x^3} \]
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Rule 211
Rule 467
Rule 1275
Rubi steps \begin{align*} \text {integral}& = \frac {b (A b-a B) x}{2 a^3 \left (a+b x^2\right )}-\frac {1}{2} b \int \frac {-\frac {2 A}{a b}+\frac {2 (A b-a B) x^2}{a^2 b}-\frac {(A b-a B) x^4}{a^3}}{x^4 \left (a+b x^2\right )} \, dx \\ & = \frac {b (A b-a B) x}{2 a^3 \left (a+b x^2\right )}-\frac {1}{2} b \int \left (-\frac {2 A}{a^2 b x^4}-\frac {2 (-2 A b+a B)}{a^3 b x^2}+\frac {-5 A b+3 a B}{a^3 \left (a+b x^2\right )}\right ) \, dx \\ & = -\frac {A}{3 a^2 x^3}+\frac {2 A b-a B}{a^3 x}+\frac {b (A b-a B) x}{2 a^3 \left (a+b x^2\right )}+\frac {(b (5 A b-3 a B)) \int \frac {1}{a+b x^2} \, dx}{2 a^3} \\ & = -\frac {A}{3 a^2 x^3}+\frac {2 A b-a B}{a^3 x}+\frac {b (A b-a B) x}{2 a^3 \left (a+b x^2\right )}+\frac {\sqrt {b} (5 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {A}{3 a^2 x^3}+\frac {2 A b-a B}{a^3 x}-\frac {b (-A b+a B) x}{2 a^3 \left (a+b x^2\right )}-\frac {\sqrt {b} (-5 A b+3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}} \]
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Time = 2.48 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {A}{3 a^{2} x^{3}}-\frac {-2 A b +B a}{x \,a^{3}}+\frac {b \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (5 A b -3 B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}\) | \(78\) |
risch | \(\frac {\frac {b \left (5 A b -3 B a \right ) x^{4}}{2 a^{3}}+\frac {\left (5 A b -3 B a \right ) x^{2}}{3 a^{2}}-\frac {A}{3 a}}{x^{3} \left (b \,x^{2}+a \right )}+\frac {5 \sqrt {-a b}\, \ln \left (-b x -\sqrt {-a b}\right ) A b}{4 a^{4}}-\frac {3 \sqrt {-a b}\, \ln \left (-b x -\sqrt {-a b}\right ) B}{4 a^{3}}-\frac {5 \sqrt {-a b}\, \ln \left (-b x +\sqrt {-a b}\right ) A b}{4 a^{4}}+\frac {3 \sqrt {-a b}\, \ln \left (-b x +\sqrt {-a b}\right ) B}{4 a^{3}}\) | \(159\) |
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Time = 0.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.78 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^2} \, dx=\left [-\frac {6 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{4} + 4 \, A a^{2} + 4 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2} + 3 \, {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{5} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{12 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, -\frac {3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2} + 3 \, {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{5} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{6 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (82) = 164\).
Time = 0.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.04 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {- \frac {b}{a^{7}}} \left (- 5 A b + 3 B a\right ) \log {\left (- \frac {a^{4} \sqrt {- \frac {b}{a^{7}}} \left (- 5 A b + 3 B a\right )}{- 5 A b^{2} + 3 B a b} + x \right )}}{4} - \frac {\sqrt {- \frac {b}{a^{7}}} \left (- 5 A b + 3 B a\right ) \log {\left (\frac {a^{4} \sqrt {- \frac {b}{a^{7}}} \left (- 5 A b + 3 B a\right )}{- 5 A b^{2} + 3 B a b} + x \right )}}{4} + \frac {- 2 A a^{2} + x^{4} \cdot \left (15 A b^{2} - 9 B a b\right ) + x^{2} \cdot \left (10 A a b - 6 B a^{2}\right )}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}}{6 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} - \frac {{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} - \frac {B a b x - A b^{2} x}{2 \, {\left (b x^{2} + a\right )} a^{3}} - \frac {3 \, B a x^{2} - 6 \, A b x^{2} + A a}{3 \, a^{3} x^{3}} \]
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Time = 4.90 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {\frac {x^2\,\left (5\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {A}{3\,a}+\frac {b\,x^4\,\left (5\,A\,b-3\,B\,a\right )}{2\,a^3}}{b\,x^5+a\,x^3}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (5\,A\,b-3\,B\,a\right )}{2\,a^{7/2}} \]
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