Integrand size = 23, antiderivative size = 131 \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} x^{1+m} \sqrt {c-a c x}}{(3+2 m) \sqrt {1-\frac {1}{a x}}}-\frac {2 (5+4 m) x^m \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-m,\frac {1}{2}-m,-\frac {1}{a x}\right )}{a (1+2 m) (3+2 m) \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.20 (sec) , antiderivative size = 131, 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 = {6311, 6316, 80, 66} \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\frac {2 \sqrt {\frac {1}{a x}+1} x^{m+1} \sqrt {c-a c x}}{(2 m+3) \sqrt {1-\frac {1}{a x}}}-\frac {2 (4 m+5) x^m \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m-\frac {1}{2},\frac {1}{2}-m,-\frac {1}{a x}\right )}{a (2 m+1) (2 m+3) \sqrt {1-\frac {1}{a x}}} \]
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Rule 66
Rule 80
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 \frac {x^{-\frac {5}{2}-m} \left (1-\frac {x}{a}\right )}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 \sqrt {1+\frac {1}{a x}} x^{1+m} \sqrt {c-a c x}}{(3+2 m) \sqrt {1-\frac {1}{a x}}}+\frac {\left ((5+4 m) \left (\frac {1}{x}\right )^{\frac {1}{2}+m} x^m \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {x^{-\frac {3}{2}-m}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a (3+2 m) \sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 \sqrt {1+\frac {1}{a x}} x^{1+m} \sqrt {c-a c x}}{(3+2 m) \sqrt {1-\frac {1}{a x}}}-\frac {2 (5+4 m) x^m \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-m,\frac {1}{2}-m,-\frac {1}{a x}\right )}{a (1+2 m) (3+2 m) \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.78 \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\frac {2 x^m \sqrt {c-a c x} \left (a (1+2 m) \sqrt {1+\frac {1}{a x}} x-(5+4 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-m,\frac {1}{2}-m,-\frac {1}{a x}\right )\right )}{a (1+2 m) (3+2 m) \sqrt {1-\frac {1}{a x}}} \]
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\[\int x^{m} \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}d x\]
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\[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\int { \sqrt {-a c x + c} x^{m} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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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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\[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\int { \sqrt {-a c x + c} x^{m} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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\[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\int { \sqrt {-a c x + c} x^{m} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a c x} \, dx=\int x^m\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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