Integrand size = 17, antiderivative size = 39 \[ \int \frac {A+B x^2}{a+b x^2} \, dx=\frac {B x}{b}+\frac {(A b-a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 39, 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 = {396, 211} \[ \int \frac {A+B x^2}{a+b x^2} \, dx=\frac {(A b-a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {B x}{b} \]
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Rule 211
Rule 396
Rubi steps \begin{align*} \text {integral}& = \frac {B x}{b}-\frac {(-A b+a B) \int \frac {1}{a+b x^2} \, dx}{b} \\ & = \frac {B x}{b}+\frac {(A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^2}{a+b x^2} \, dx=\frac {B x}{b}-\frac {(-A b+a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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Time = 2.47 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {B x}{b}+\frac {\left (A b -B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(34\) |
risch | \(\frac {B x}{b}-\frac {\ln \left (b x +\sqrt {-a b}\right ) A}{2 \sqrt {-a b}}+\frac {\ln \left (b x +\sqrt {-a b}\right ) B a}{2 b \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) A}{2 \sqrt {-a b}}-\frac {\ln \left (-b x +\sqrt {-a b}\right ) B a}{2 b \sqrt {-a b}}\) | \(98\) |
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Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.54 \[ \int \frac {A+B x^2}{a+b x^2} \, dx=\left [\frac {2 \, B a b x + {\left (B a - A b\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b^{2}}, \frac {B a b x - {\left (B a - A b\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a b^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (34) = 68\).
Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.10 \[ \int \frac {A+B x^2}{a+b x^2} \, dx=\frac {B x}{b} + \frac {\sqrt {- \frac {1}{a b^{3}}} \left (- A b + B a\right ) \log {\left (- a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{a b^{3}}} \left (- A b + B a\right ) \log {\left (a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{2} \]
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Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^2}{a+b x^2} \, dx=\frac {B x}{b} - \frac {{\left (B a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^2}{a+b x^2} \, dx=\frac {B x}{b} - \frac {{\left (B a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} \]
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Time = 4.92 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x^2}{a+b x^2} \, dx=\frac {B\,x}{b}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b-B\,a\right )}{\sqrt {a}\,b^{3/2}} \]
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