Integrand size = 21, antiderivative size = 55 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {375, 272, 52, 65, 214} \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right ) \]
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Rule 52
Rule 65
Rule 214
Rule 272
Rule 375
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {a+b x^3}}{x} \, dx,x,\sqrt {c x^2}\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\left (c x^2\right )^{3/2}\right ) \\ & = \frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {1}{3} a \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\left (c x^2\right )^{3/2}\right ) \\ & = \frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \left (c x^2\right )^{3/2}}\right )}{3 b} \\ & = \frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\frac {2}{3} \left (\sqrt {a+b \left (c x^2\right )^{3/2}}-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )\right ) \]
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\[\int \frac {\sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{x}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.42 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\left [\frac {1}{3} \, \sqrt {a} \log \left (\frac {b c^{2} x^{4} - 2 \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a} \sqrt {c x^{2}} \sqrt {a} + 2 \, \sqrt {c x^{2}} a}{x^{4}}\right ) + \frac {2}{3} \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a}, \frac {2}{3} \, \sqrt {-a} \arctan \left (\frac {\sqrt {\sqrt {c x^{2}} b c x^{2} + a} \sqrt {-a}}{a}\right ) + \frac {2}{3} \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a}\right ] \]
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\[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\int \frac {\sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}}{x}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\frac {1}{3} \, \sqrt {a} \log \left (\frac {\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} - \sqrt {a}}{\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} + \sqrt {a}}\right ) + \frac {2}{3} \, \sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} \]
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Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\frac {2 \, a \arctan \left (\frac {\sqrt {b c^{\frac {3}{2}} x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} + \frac {2}{3} \, \sqrt {b c^{\frac {3}{2}} x^{3} + a} \]
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Timed out. \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\int \frac {\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}}}{x} \,d x \]
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