Integrand size = 17, antiderivative size = 44 \[ \int \sqrt {1+\frac {b}{x^2}} (c x)^m \, dx=\frac {(c x)^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-1-m),\frac {1-m}{2},-\frac {b}{x^2}\right )}{c (1+m)} \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {346, 371} \[ \int \sqrt {1+\frac {b}{x^2}} (c x)^m \, dx=\frac {(c x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-m-1),\frac {1-m}{2},-\frac {b}{x^2}\right )}{c (m+1)} \]
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Rule 346
Rule 371
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\left (\frac {1}{x}\right )^{1+m} (c x)^{1+m}\right ) \text {Subst}\left (\int x^{-2-m} \sqrt {1+b x^2} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {(c x)^{1+m} \, _2F_1\left (-\frac {1}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};-\frac {b}{x^2}\right )}{c (1+m)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.32 \[ \int \sqrt {1+\frac {b}{x^2}} (c x)^m \, dx=\frac {\sqrt {1+\frac {b}{x^2}} x (c x)^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {m}{2},1+\frac {m}{2},-\frac {x^2}{b}\right )}{m \sqrt {1+\frac {x^2}{b}}} \]
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\[\int \sqrt {1+\frac {b}{x^{2}}}\, \left (c x \right )^{m}d x\]
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\[ \int \sqrt {1+\frac {b}{x^2}} (c x)^m \, dx=\int { \left (c x\right )^{m} \sqrt {\frac {b}{x^{2}} + 1} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.27 \[ \int \sqrt {1+\frac {b}{x^2}} (c x)^m \, dx=- \frac {b^{- \frac {m}{2}} b^{\frac {m}{2} + \frac {1}{2}} c^{m} x^{m} \Gamma \left (- \frac {m}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} \\ \frac {m}{2} + 1 \end {matrix}\middle | {\frac {x^{2} e^{i \pi }}{b}} \right )}}{2 \Gamma \left (1 - \frac {m}{2}\right )} \]
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\[ \int \sqrt {1+\frac {b}{x^2}} (c x)^m \, dx=\int { \left (c x\right )^{m} \sqrt {\frac {b}{x^{2}} + 1} \,d x } \]
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\[ \int \sqrt {1+\frac {b}{x^2}} (c x)^m \, dx=\int { \left (c x\right )^{m} \sqrt {\frac {b}{x^{2}} + 1} \,d x } \]
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Timed out. \[ \int \sqrt {1+\frac {b}{x^2}} (c x)^m \, dx=\int {\left (c\,x\right )}^m\,\sqrt {\frac {b}{x^2}+1} \,d x \]
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