Integrand size = 24, antiderivative size = 49 \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=-\frac {\text {arctanh}\left (\frac {x^{3/2} (2 a+b x)}{2 \sqrt {a} \sqrt {a x^3+b x^4+c x^5}}\right )}{\sqrt {a}} \]
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Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2022, 1927, 212} \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=-\frac {\text {arctanh}\left (\frac {x^{3/2} (2 a+b x)}{2 \sqrt {a} \sqrt {a x^3+b x^4+c x^5}}\right )}{\sqrt {a}} \]
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Rule 212
Rule 1927
Rule 2022
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {x}}{\sqrt {a x^3+b x^4+c x^5}} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x^{3/2} (2 a+b x)}{\sqrt {a x^3+b x^4+c x^5}}\right )\right ) \\ & = -\frac {\tanh ^{-1}\left (\frac {x^{3/2} (2 a+b x)}{2 \sqrt {a} \sqrt {a x^3+b x^4+c x^5}}\right )}{\sqrt {a}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\frac {2 x^{3/2} \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {x^3 (a+x (b+c x))}} \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.35
method | result | size |
default | \(-\frac {x^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {x^{3} \left (c \,x^{2}+b x +a \right )}\, \sqrt {a}}\) | \(66\) |
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Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.84 \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\left [\frac {\log \left (\frac {8 \, a b x^{3} + {\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a^{2} x^{2} - 4 \, \sqrt {c x^{5} + b x^{4} + a x^{3}} {\left (b x + 2 \, a\right )} \sqrt {a} \sqrt {x}}{x^{4}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{5} + b x^{4} + a x^{3}} {\left (b x + 2 \, a\right )} \sqrt {-a} \sqrt {x}}{2 \, {\left (a c x^{4} + a b x^{3} + a^{2} x^{2}\right )}}\right )}{a}\right ] \]
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Timed out. \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {{\left (c x^{2} + b x + a\right )} x^{3}}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\frac {2 \, {\left (\frac {\arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right )}{\sqrt {-a}}\right )}}{\mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\int \frac {\sqrt {x}}{\sqrt {x^3\,\left (c\,x^2+b\,x+a\right )}} \,d x \]
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