Integrand size = 20, antiderivative size = 98 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\frac {2}{5} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (-3+n)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{2} (-3+n),-\frac {3}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6311, 6316, 134} \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\frac {2}{5} x (c-a c x)^{3/2} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {n-3}{2},-\frac {3}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \]
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Rule 134
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {(c-a c x)^{3/2} \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}} \\ & = \frac {2}{5} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (-3+n)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{2} (-3+n),-\frac {3}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.03 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 c \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {-1+a x}{1+a x}\right )^{\frac {1}{2} (-1+n)} (1+a x)^2 \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{2} (-3+n),-\frac {3}{2},\frac {2}{1+a x}\right )}{5 a} \]
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\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{\frac {3}{2}}d x\]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\int { {\left (-a c x + c\right )}^{\frac {3}{2}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\int \left (- c \left (a x - 1\right )\right )^{\frac {3}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\int { {\left (-a c x + c\right )}^{\frac {3}{2}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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Exception generated. \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^{3/2} \,d x \]
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