Integrand size = 27, antiderivative size = 137 \[ \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=-\frac {3 x^m \sqrt {c-a^2 c x^2}}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x^m \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x)}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6327, 6328, 90, 66} \[ \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {4 x^m \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}(1,m+1,m+2,-a x)}{a (m+1) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 x^m \sqrt {c-a^2 c x^2}}{a (m+1) \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 66
Rule 90
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^{1+m} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \frac {x^m (-1+a x)^2}{1+a x} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \left (-3 x^m+a x^{1+m}+\frac {4 x^m}{1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = -\frac {3 x^m \sqrt {c-a^2 c x^2}}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\left (4 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^m}{1+a x} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = -\frac {3 x^m \sqrt {c-a^2 c x^2}}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x^m \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x)}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.55 \[ \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^m \sqrt {c-a^2 c x^2} (-6+a x+m (-3+a x)+4 (2+m) \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x))}{a (1+m) (2+m) \sqrt {1-\frac {1}{a^2 x^2}}} \]
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\[\int x^{m} \sqrt {-a^{2} c \,x^{2}+c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}d x\]
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\[ \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} x^{m} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\text {Timed out} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} x^{m} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int x^m\,\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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