Integrand size = 18, antiderivative size = 43 \[ \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c}} \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1121, 635, 212} \[ \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c}} \]
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Rule 212
Rule 635
Rule 1121
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right ) \\ & = \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right ) \\ & = \frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx=-\frac {\log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{2 \sqrt {c}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}\) | \(35\) |
elliptic | \(\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}\) | \(35\) |
pseudoelliptic | \(\frac {-\ln \left (2\right )+\ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )}{2 \sqrt {c}}\) | \(43\) |
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none
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.74 \[ \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx=\left [\frac {\log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right )}{4 \, \sqrt {c}}, -\frac {\sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right )}{2 \, c}\right ] \]
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\[ \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {x}{\sqrt {a + b x^{2} + c x^{4}}}\, dx \]
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Exception generated. \[ \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx=-\frac {\log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{2 \, \sqrt {c}} \]
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Time = 13.37 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )}{2\,\sqrt {c}} \]
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