Integrand size = 18, antiderivative size = 38 \[ \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a}} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {738, 212} \[ \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a}} \]
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Rule 212
Rule 738
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )\right ) \\ & = -\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Time = 1.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}\) | \(35\) |
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none
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.92 \[ \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx=\left [\frac {\log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right )}{a}\right ] \]
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\[ \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{x \sqrt {a + b x + c x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx=\frac {2 \, \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} \]
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Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx=-\frac {\ln \left (\frac {b}{2}+\frac {a}{x}+\frac {\sqrt {a}\,\sqrt {c\,x^2+b\,x+a}}{x}\right )}{\sqrt {a}} \]
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