Integrand size = 28, antiderivative size = 110 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {A}{a^2 x}+\frac {b B-a D-b \left (\frac {A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}-\frac {(3 A b-a C) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b}}+\frac {B \log (x)}{a^2}-\frac {B \log \left (a+b x^2\right )}{2 a^2} \]
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Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1819, 1816, 649, 211, 266} \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {(3 A b-a C) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b}}-\frac {A}{a^2 x}-\frac {B \log \left (a+b x^2\right )}{2 a^2}+\frac {B \log (x)}{a^2}+\frac {-b x \left (\frac {A b}{a}-C\right )-a D+b B}{2 a b \left (a+b x^2\right )} \]
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Rule 211
Rule 266
Rule 649
Rule 1816
Rule 1819
Rubi steps \begin{align*} \text {integral}& = \frac {b B-a D-b \left (\frac {A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}-\frac {\int \frac {-2 A-2 B x+\left (\frac {A b}{a}-C\right ) x^2}{x^2 \left (a+b x^2\right )} \, dx}{2 a} \\ & = \frac {b B-a D-b \left (\frac {A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}-\frac {\int \left (-\frac {2 A}{a x^2}-\frac {2 B}{a x}+\frac {3 A b-a C+2 b B x}{a \left (a+b x^2\right )}\right ) \, dx}{2 a} \\ & = -\frac {A}{a^2 x}+\frac {b B-a D-b \left (\frac {A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}+\frac {B \log (x)}{a^2}-\frac {\int \frac {3 A b-a C+2 b B x}{a+b x^2} \, dx}{2 a^2} \\ & = -\frac {A}{a^2 x}+\frac {b B-a D-b \left (\frac {A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}+\frac {B \log (x)}{a^2}-\frac {(b B) \int \frac {x}{a+b x^2} \, dx}{a^2}-\frac {(3 A b-a C) \int \frac {1}{a+b x^2} \, dx}{2 a^2} \\ & = -\frac {A}{a^2 x}+\frac {b B-a D-b \left (\frac {A b}{a}-C\right ) x}{2 a b \left (a+b x^2\right )}-\frac {(3 A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b}}+\frac {B \log (x)}{a^2}-\frac {B \log \left (a+b x^2\right )}{2 a^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {A}{a^2 x}+\frac {a b B-a^2 D-A b^2 x+a b C x}{2 a^2 b \left (a+b x^2\right )}+\frac {(-3 A b+a C) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b}}+\frac {B \log (x)}{a^2}-\frac {B \log \left (a+b x^2\right )}{2 a^2} \]
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Time = 3.55 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {A}{a^{2} x}+\frac {B \ln \left (x \right )}{a^{2}}-\frac {\frac {\left (\frac {A b}{2}-\frac {C a}{2}\right ) x -\frac {a \left (B b -D a \right )}{2 b}}{b \,x^{2}+a}+\frac {B \ln \left (b \,x^{2}+a \right )}{2}+\frac {\left (3 A b -C a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{a^{2}}\) | \(96\) |
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Time = 0.30 (sec) , antiderivative size = 336, normalized size of antiderivative = 3.05 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^2} \, dx=\left [-\frac {4 \, A a^{2} b - 2 \, {\left (C a^{2} b - 3 \, A a b^{2}\right )} x^{2} - {\left ({\left (C a b - 3 \, A b^{2}\right )} x^{3} + {\left (C a^{2} - 3 \, A a b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (D a^{3} - B a^{2} b\right )} x + 2 \, {\left (B a b^{2} x^{3} + B a^{2} b x\right )} \log \left (b x^{2} + a\right ) - 4 \, {\left (B a b^{2} x^{3} + B a^{2} b x\right )} \log \left (x\right )}{4 \, {\left (a^{3} b^{2} x^{3} + a^{4} b x\right )}}, -\frac {2 \, A a^{2} b - {\left (C a^{2} b - 3 \, A a b^{2}\right )} x^{2} - {\left ({\left (C a b - 3 \, A b^{2}\right )} x^{3} + {\left (C a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (D a^{3} - B a^{2} b\right )} x + {\left (B a b^{2} x^{3} + B a^{2} b x\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (B a b^{2} x^{3} + B a^{2} b x\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{2} x^{3} + a^{4} b x\right )}}\right ] \]
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Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {2 \, A a b - {\left (C a b - 3 \, A b^{2}\right )} x^{2} + {\left (D a^{2} - B a b\right )} x}{2 \, {\left (a^{2} b^{2} x^{3} + a^{3} b x\right )}} - \frac {B \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {B \log \left (x\right )}{a^{2}} + \frac {{\left (C a - 3 \, A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {B \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {B \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {{\left (C a - 3 \, A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} + \frac {C a b x^{2} - 3 \, A b^{2} x^{2} - D a^{2} x + B a b x - 2 \, A a b}{2 \, {\left (b x^{3} + a x\right )} a^{2} b} \]
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Time = 6.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {B}{2\,a\,\left (b\,x^2+a\right )}-\frac {\frac {A}{a}+\frac {3\,A\,b\,x^2}{2\,a^2}}{b\,x^3+a\,x}-\frac {B\,\ln \left (b\,x^2+a\right )}{2\,a^2}+\frac {B\,\ln \left (x\right )}{a^2}-\frac {D}{2\,b\,\left (b\,x^2+a\right )}+\frac {C\,x}{2\,a\,\left (b\,x^2+a\right )}-\frac {3\,A\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{5/2}}+\frac {C\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}} \]
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