Integrand size = 23, antiderivative size = 87 \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=-\frac {2 b (d x)^{1+m} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m \left (a+\frac {b}{\sqrt {c x^2}}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},2+m,\frac {5}{2},1+\frac {b}{a \sqrt {c x^2}}\right )}{3 a^2 d \sqrt {c x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {375, 346, 69, 67} \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=-\frac {2 b (d x)^{m+1} \left (a+\frac {b}{\sqrt {c x^2}}\right )^{3/2} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m \operatorname {Hypergeometric2F1}\left (\frac {3}{2},m+2,\frac {5}{2},\frac {b}{a \sqrt {c x^2}}+1\right )}{3 a^2 d \sqrt {c x^2}} \]
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Rule 67
Rule 69
Rule 346
Rule 375
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \text {Subst}\left (\int \sqrt {a+\frac {b}{x}} x^m \, dx,x,\sqrt {c x^2}\right )}{d} \\ & = -\frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \text {Subst}\left (\int x^{-2-m} \sqrt {a+b x} \, dx,x,\frac {1}{\sqrt {c x^2}}\right )}{d} \\ & = -\frac {\left (b^2 (d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)+\frac {m}{2}} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m\right ) \text {Subst}\left (\int \left (-\frac {b x}{a}\right )^{-2-m} \sqrt {a+b x} \, dx,x,\frac {1}{\sqrt {c x^2}}\right )}{a^2 d} \\ & = -\frac {2 b (d x)^{1+m} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m \left (a+\frac {b}{\sqrt {c x^2}}\right )^{3/2} \, _2F_1\left (\frac {3}{2},2+m;\frac {5}{2};1+\frac {b}{a \sqrt {c x^2}}\right )}{3 a^2 d \sqrt {c x^2}} \\ \end{align*}
Time = 1.40 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\frac {2 x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}+m,\frac {3}{2}+m,-\frac {a \sqrt {c x^2}}{b}\right )}{(1+2 m) \sqrt {1+\frac {a \sqrt {c x^2}}{b}}} \]
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\[\int \left (d x \right )^{m} \sqrt {a +\frac {b}{\sqrt {c \,x^{2}}}}d x\]
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\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}} \,d x } \]
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\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}}\, dx \]
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\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}} \,d x } \]
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\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}} \,d x } \]
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Timed out. \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{\sqrt {c\,x^2}}} \,d x \]
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