Integrand size = 24, antiderivative size = 51 \[ \int \frac {1}{\sqrt {x} \sqrt {a x+b x^3+c x^5}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x+b x^3+c x^5}}\right )}{2 \sqrt {a}} \]
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Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1927, 212} \[ \int \frac {1}{\sqrt {x} \sqrt {a x+b x^3+c x^5}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x+b x^3+c x^5}}\right )}{2 \sqrt {a}} \]
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Rule 212
Rule 1927
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {\sqrt {x} \left (2 a+b x^2\right )}{\sqrt {a x+b x^3+c x^5}}\right ) \\ & = -\frac {\tanh ^{-1}\left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x+b x^3+c x^5}}\right )}{2 \sqrt {a}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\sqrt {x} \sqrt {a x+b x^3+c x^5}} \, dx=\frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {x \left (a+b x^2+c x^4\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.41
method | result | size |
default | \(-\frac {\sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}\, \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2 \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}\) | \(72\) |
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Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.69 \[ \int \frac {1}{\sqrt {x} \sqrt {a x+b x^3+c x^5}} \, dx=\left [\frac {\log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} \sqrt {x}}{x^{5}}\right )}{4 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{5} + b x^{3} + a x} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a} \sqrt {x}}{2 \, {\left (a c x^{5} + a b x^{3} + a^{2} x\right )}}\right )}{2 \, a}\right ] \]
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\[ \int \frac {1}{\sqrt {x} \sqrt {a x+b x^3+c x^5}} \, dx=\int \frac {1}{\sqrt {x} \sqrt {x \left (a + b x^{2} + c x^{4}\right )}}\, dx \]
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\[ \int \frac {1}{\sqrt {x} \sqrt {a x+b x^3+c x^5}} \, dx=\int { \frac {1}{\sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}} \,d x } \]
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Time = 0.43 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {x} \sqrt {a x+b x^3+c x^5}} \, dx=\frac {\arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right )}{\sqrt {-a}} \]
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Timed out. \[ \int \frac {1}{\sqrt {x} \sqrt {a x+b x^3+c x^5}} \, dx=\int \frac {1}{\sqrt {x}\,\sqrt {c\,x^5+b\,x^3+a\,x}} \,d x \]
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