Integrand size = 17, antiderivative size = 63 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^2} \, dx=\frac {(A b-a B) x}{2 a b \left (a+b x^2\right )}+\frac {(A b+a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {393, 211} \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^2} \, dx=\frac {(a B+A b) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {x (A b-a B)}{2 a b \left (a+b x^2\right )} \]
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Rule 211
Rule 393
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) x}{2 a b \left (a+b x^2\right )}+\frac {(A b+a B) \int \frac {1}{a+b x^2} \, dx}{2 a b} \\ & = \frac {(A b-a B) x}{2 a b \left (a+b x^2\right )}+\frac {(A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^2} \, dx=-\frac {(-A b+a B) x}{2 a b \left (a+b x^2\right )}+\frac {(A b+a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}} \]
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Time = 2.52 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\left (A b -B a \right ) x}{2 a b \left (b \,x^{2}+a \right )}+\frac {\left (A b +B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a b \sqrt {a b}}\) | \(57\) |
risch | \(\frac {\left (A b -B a \right ) x}{2 a b \left (b \,x^{2}+a \right )}-\frac {\ln \left (b x +\sqrt {-a b}\right ) A}{4 \sqrt {-a b}\, a}-\frac {\ln \left (b x +\sqrt {-a b}\right ) B}{4 \sqrt {-a b}\, b}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) A}{4 \sqrt {-a b}\, a}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) B}{4 \sqrt {-a b}\, b}\) | \(122\) |
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Time = 0.28 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.89 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^2} \, dx=\left [-\frac {{\left (B a^{2} + A a b + {\left (B a b + A b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (B a^{2} b - A a b^{2}\right )} x}{4 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}, \frac {{\left (B a^{2} + A a b + {\left (B a b + A b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (B a^{2} b - A a b^{2}\right )} x}{2 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (54) = 108\).
Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.78 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^2} \, dx=\frac {x \left (A b - B a\right )}{2 a^{2} b + 2 a b^{2} x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b^{3}}} \left (A b + B a\right ) \log {\left (- a^{2} b \sqrt {- \frac {1}{a^{3} b^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b^{3}}} \left (A b + B a\right ) \log {\left (a^{2} b \sqrt {- \frac {1}{a^{3} b^{3}}} + x \right )}}{4} \]
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Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (B a - A b\right )} x}{2 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} + \frac {{\left (B a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} \]
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Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (B a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} - \frac {B a x - A b x}{2 \, {\left (b x^{2} + a\right )} a b} \]
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Time = 5.15 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x^2}{\left (a+b x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b+B\,a\right )}{2\,a^{3/2}\,b^{3/2}}+\frac {x\,\left (A\,b-B\,a\right )}{2\,a\,b\,\left (b\,x^2+a\right )} \]
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