Integrand size = 27, antiderivative size = 944 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {3 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{(3+n) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {2^{\frac {5+n}{2}} a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-n),\frac {1}{2} (-3-n),\frac {1}{2} (-1-n),\frac {a-\frac {1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}} \]
[Out]
Time = 0.47 (sec) , antiderivative size = 944, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6327, 6330, 128, 47, 37, 71} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {2^{\frac {n+5}{2}} a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-3),\frac {1}{2} (-n-3),\frac {1}{2} (-n-1),\frac {a-\frac {1}{x}}{2 a}\right ) \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{\frac {n-3}{2}} x^5 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{\frac {n-1}{2}} x^5 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{\frac {n+1}{2}} x^5 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{\frac {n+3}{2}} x^5 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac {3 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{\frac {n-3}{2}} x^5 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{\frac {n-1}{2}} x^5 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{\frac {n+1}{2}} x^5 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{\frac {n-3}{2}} x^5 \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{\frac {n-1}{2}} x^5 \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{\frac {n-3}{2}} x^5 \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}} \]
[In]
[Out]
Rule 37
Rule 47
Rule 71
Rule 128
Rule 6327
Rule 6330
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^6} \, dx}{\left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int x^4 \left (1-\frac {x}{a}\right )^{-\frac {5}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {5}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (a^4 \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-5-n)} \left (1+\frac {x}{a}\right )^{-\frac {5}{2}+\frac {n}{2}}-4 a^4 \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-5-n)} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}}+6 a^4 \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-5-n)} \left (1+\frac {x}{a}\right )^{-\frac {1}{2}+\frac {n}{2}}-4 a^4 \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-5-n)} \left (1+\frac {x}{a}\right )^{\frac {1}{2}+\frac {n}{2}}+a^4 \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-5-n)} \left (1+\frac {x}{a}\right )^{\frac {3}{2}+\frac {n}{2}}\right ) \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {\left (a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-5-n)} \left (1+\frac {x}{a}\right )^{-\frac {5}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-5-n)} \left (1+\frac {x}{a}\right )^{\frac {3}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (4 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-5-n)} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (4 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-5-n)} \left (1+\frac {x}{a}\right )^{\frac {1}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (6 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-5-n)} \left (1+\frac {x}{a}\right )^{-\frac {1}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {2^{\frac {5+n}{2}} a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-n),\frac {1}{2} (-3-n),\frac {1}{2} (-1-n),\frac {a-\frac {1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (3 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {x}{a}\right )^{-\frac {5}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (6 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {x}{a}\right )^{-\frac {1}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {3 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {2^{\frac {5+n}{2}} a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-n),\frac {1}{2} (-3-n),\frac {1}{2} (-1-n),\frac {a-\frac {1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (6 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {x}{a}\right )^{-\frac {5}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {3 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {2^{\frac {5+n}{2}} a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-n),\frac {1}{2} (-3-n),\frac {1}{2} (-1-n),\frac {a-\frac {1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (6 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{\frac {1-n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {5}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {3 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {2^{\frac {5+n}{2}} a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-n),\frac {1}{2} (-3-n),\frac {1}{2} (-1-n),\frac {a-\frac {1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}} \\ \end{align*}
Time = 2.22 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.23 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (42-2 n^2-45 a n x+5 a n^3 x\right )+6 a \left (-1+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (2 \coth ^{-1}(a x)\right )-n \left (-1+n^2\right ) \left (-1+a^2 x^2\right ) \cosh \left (3 \coth ^{-1}(a x)\right )\right )-8 e^{(1+n) \coth ^{-1}(a x)} \left (9-9 n-n^2+n^3\right ) \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-e^{2 \coth ^{-1}(a x)}\right )}{4 a c^2 (-1+n) (1+n) \left (-9+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a^2 c x^2}} \]
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\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]
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Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]
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