Integrand size = 21, antiderivative size = 65 \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\frac {2 x^{1+m} \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2}-m,-\frac {1}{2}-m,-\frac {1}{a x}\right )}{(3+2 m) \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6311, 6316, 66} \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\frac {2 x^{m+1} \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-m-\frac {3}{2},-m-\frac {1}{2},-\frac {1}{a x}\right )}{(2 m+3) \sqrt {1-\frac {1}{a x}}} \]
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Rule 66
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a c x} \int e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^{\frac {1}{2}+m} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{\frac {1}{2}+m} x^m \sqrt {c-a c x}\right ) \text {Subst}\left (\int x^{-\frac {5}{2}-m} \sqrt {1+\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 x^{1+m} \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2}-m,-\frac {1}{2}-m,-\frac {1}{a x}\right )}{(3+2 m) \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03 \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=-\frac {x^{1+m} \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2}-m,-\frac {1}{2}-m,-\frac {1}{a x}\right )}{\left (-\frac {3}{2}-m\right ) \sqrt {1-\frac {1}{a x}}} \]
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\[\int \frac {x^{m} \sqrt {-a c x +c}}{\sqrt {\frac {a x -1}{a x +1}}}d x\]
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\[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\int { \frac {\sqrt {-a c x + c} x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\text {Timed out} \]
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\[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\int { \frac {\sqrt {-a c x + c} x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\int { \frac {\sqrt {-a c x + c} x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\int \frac {x^m\,\sqrt {c-a\,c\,x}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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