Integrand size = 24, antiderivative size = 45 \[ \int \frac {x^2}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {x}{2 b \left (a+b x^2\right )}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 294, 211} \[ \int \frac {x^2}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {x}{2 b \left (a+b x^2\right )} \]
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Rule 28
Rule 211
Rule 294
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {x^2}{\left (a b+b^2 x^2\right )^2} \, dx \\ & = -\frac {x}{2 b \left (a+b x^2\right )}+\frac {1}{2} \int \frac {1}{a b+b^2 x^2} \, dx \\ & = -\frac {x}{2 b \left (a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {x}{2 b \left (a+b x^2\right )}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {x}{2 b \left (b \,x^{2}+a \right )}+\frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 b \sqrt {a b}}\) | \(36\) |
risch | \(-\frac {x}{2 b \left (b \,x^{2}+a \right )}-\frac {\ln \left (b x +\sqrt {-a b}\right )}{4 \sqrt {-a b}\, b}+\frac {\ln \left (-b x +\sqrt {-a b}\right )}{4 \sqrt {-a b}\, b}\) | \(62\) |
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Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.67 \[ \int \frac {x^2}{a^2+2 a b x^2+b^2 x^4} \, dx=\left [-\frac {2 \, a b x + {\left (b x^{2} + a\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{4 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}}, -\frac {a b x - {\left (b x^{2} + a\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (36) = 72\).
Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.73 \[ \int \frac {x^2}{a^2+2 a b x^2+b^2 x^4} \, dx=- \frac {x}{2 a b + 2 b^{2} x^{2}} - \frac {\sqrt {- \frac {1}{a b^{3}}} \log {\left (- a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a b^{3}}} \log {\left (a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{4} \]
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Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {x}{2 \, {\left (b^{2} x^{2} + a b\right )}} + \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b} \]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b} - \frac {x}{2 \, {\left (b x^{2} + a\right )} b} \]
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Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,b^{3/2}}-\frac {x}{2\,b\,\left (b\,x^2+a\right )} \]
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