Integrand size = 17, antiderivative size = 39 \[ \int \frac {c+d x^2}{a+b x^2} \, dx=\frac {d x}{b}+\frac {(b c-a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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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} \[ \int \frac {c+d x^2}{a+b x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)}{\sqrt {a} b^{3/2}}+\frac {d x}{b} \]
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Rule 211
Rule 396
Rubi steps \begin{align*} \text {integral}& = \frac {d x}{b}-\frac {(-b c+a d) \int \frac {1}{a+b x^2} \, dx}{b} \\ & = \frac {d x}{b}+\frac {(b c-a d) \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 \[ \int \frac {c+d x^2}{a+b x^2} \, dx=\frac {d x}{b}-\frac {(-b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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Time = 2.62 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {d x}{b}+\frac {\left (-a d +b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(34\) |
risch | \(\frac {d x}{b}-\frac {\ln \left (b x -\sqrt {-a b}\right ) a d}{2 b \sqrt {-a b}}+\frac {\ln \left (b x -\sqrt {-a b}\right ) c}{2 \sqrt {-a b}}+\frac {\ln \left (-b x -\sqrt {-a b}\right ) a d}{2 b \sqrt {-a b}}-\frac {\ln \left (-b x -\sqrt {-a b}\right ) c}{2 \sqrt {-a b}}\) | \(106\) |
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Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.51 \[ \int \frac {c+d x^2}{a+b x^2} \, dx=\left [\frac {2 \, a b d x + \sqrt {-a b} {\left (b c - a d\right )} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b^{2}}, \frac {a b d x + \sqrt {a b} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a b^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (34) = 68\).
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.10 \[ \int \frac {c+d x^2}{a+b x^2} \, dx=\frac {\sqrt {- \frac {1}{a b^{3}}} \left (a d - b c\right ) \log {\left (- a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{a b^{3}}} \left (a d - b c\right ) \log {\left (a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{2} + \frac {d x}{b} \]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {c+d x^2}{a+b x^2} \, dx=\frac {d x}{b} + \frac {{\left (b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} \]
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {c+d x^2}{a+b x^2} \, dx=\frac {d x}{b} + \frac {{\left (b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} \]
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Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {c+d x^2}{a+b x^2} \, dx=\frac {d\,x}{b}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )}{\sqrt {a}\,b^{3/2}} \]
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