Integrand size = 15, antiderivative size = 42 \[ \int \frac {A+B x}{a+c x^2} \, dx=\frac {A \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {B \log \left (a+c x^2\right )}{2 c} \]
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Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {649, 211, 266} \[ \int \frac {A+B x}{a+c x^2} \, dx=\frac {A \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {B \log \left (a+c x^2\right )}{2 c} \]
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Rule 211
Rule 266
Rule 649
Rubi steps \begin{align*} \text {integral}& = A \int \frac {1}{a+c x^2} \, dx+B \int \frac {x}{a+c x^2} \, dx \\ & = \frac {A \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {B \log \left (a+c x^2\right )}{2 c} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{a+c x^2} \, dx=\frac {A \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {B \log \left (a+c x^2\right )}{2 c} \]
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Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {B \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {A \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}\) | \(32\) |
risch | \(\frac {\ln \left (-\sqrt {-a c}\, x +a \right ) A \sqrt {-a c}}{2 a c}+\frac {\ln \left (-\sqrt {-a c}\, x +a \right ) B}{2 c}-\frac {\ln \left (\sqrt {-a c}\, x +a \right ) A \sqrt {-a c}}{2 a c}+\frac {\ln \left (\sqrt {-a c}\, x +a \right ) B}{2 c}\) | \(90\) |
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Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.33 \[ \int \frac {A+B x}{a+c x^2} \, dx=\left [\frac {B a \log \left (c x^{2} + a\right ) - \sqrt {-a c} A \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{2 \, a c}, \frac {B a \log \left (c x^{2} + a\right ) + 2 \, \sqrt {a c} A \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{2 \, a c}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (37) = 74\).
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.95 \[ \int \frac {A+B x}{a+c x^2} \, dx=\left (- \frac {A \sqrt {- a c^{3}}}{2 a c^{2}} + \frac {B}{2 c}\right ) \log {\left (x + \frac {- B a + 2 a c \left (- \frac {A \sqrt {- a c^{3}}}{2 a c^{2}} + \frac {B}{2 c}\right )}{A c} \right )} + \left (\frac {A \sqrt {- a c^{3}}}{2 a c^{2}} + \frac {B}{2 c}\right ) \log {\left (x + \frac {- B a + 2 a c \left (\frac {A \sqrt {- a c^{3}}}{2 a c^{2}} + \frac {B}{2 c}\right )}{A c} \right )} \]
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Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x}{a+c x^2} \, dx=\frac {A \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} + \frac {B \log \left (c x^{2} + a\right )}{2 \, c} \]
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x}{a+c x^2} \, dx=\frac {A \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} + \frac {B \log \left (c x^{2} + a\right )}{2 \, c} \]
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Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x}{a+c x^2} \, dx=\frac {B\,\ln \left (c\,x^2+a\right )}{2\,c}+\frac {A\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {c}} \]
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