Integrand size = 23, antiderivative size = 84 \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\frac {x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{3} (1+m),\frac {1}{3} (1-2 m),-\frac {b}{a \sqrt {c x^3}}\right )}{(1+m) \sqrt {1+\frac {b}{a \sqrt {c x^3}}}} \]
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Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {376, 350, 348, 346, 372, 371} \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\frac {x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{3} (m+1),\frac {1}{3} (1-2 m),-\frac {b}{a \sqrt {c x^3}}\right )}{(m+1) \sqrt {\frac {b}{a \sqrt {c x^3}}+1}} \]
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Rule 346
Rule 348
Rule 350
Rule 371
Rule 372
Rule 376
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \sqrt {a+\frac {b}{\sqrt {c} x^{3/2}}} (d x)^m \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (\left (x^{-m} (d x)^m\right ) \int \sqrt {a+\frac {b}{\sqrt {c} x^{3/2}}} x^m \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (\left (2 x^{-m} (d x)^m\right ) \text {Subst}\left (\int \sqrt {a+\frac {b}{\sqrt {c} x^3}} x^{-1+2 (1+m)} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\text {Subst}\left (\left (2 x^{-m} (d x)^m\right ) \text {Subst}\left (\int x^{-1-2 (1+m)} \sqrt {a+\frac {b x^3}{\sqrt {c}}} \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\text {Subst}\left (\frac {\left (2 \sqrt {a+\frac {b}{\sqrt {c} x^{3/2}}} x^{-m} (d x)^m\right ) \text {Subst}\left (\int x^{-1-2 (1+m)} \sqrt {1+\frac {b x^3}{a \sqrt {c}}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{\sqrt {1+\frac {b}{a \sqrt {c} x^{3/2}}}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \frac {x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, _2F_1\left (-\frac {1}{2},-\frac {2}{3} (1+m);\frac {1}{3} (1-2 m);-\frac {b}{a \sqrt {c x^3}}\right )}{(1+m) \sqrt {1+\frac {b}{a \sqrt {c x^3}}}} \\ \end{align*}
Time = 1.57 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.06 \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\frac {4 x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{6}+\frac {2 m}{3},\frac {7}{6}+\frac {2 m}{3},-\frac {a \sqrt {c x^3}}{b}\right )}{(1+4 m) \sqrt {1+\frac {a \sqrt {c x^3}}{b}}} \]
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\[\int \left (d x \right )^{m} \sqrt {a +\frac {b}{\sqrt {c \,x^{3}}}}d x\]
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Exception generated. \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{3}}}}\, dx \]
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\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{3}}}} \,d x } \]
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\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{3}}}} \,d x } \]
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Timed out. \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{\sqrt {c\,x^3}}} \,d x \]
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