Integrand size = 27, antiderivative size = 463 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {(6+n) \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (15+6 n+n^2\right ) \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (18+7 n-2 n^2-n^3\right ) \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{(1-n) \left (c-a^2 c x^2\right )^{5/2}} \]
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Time = 0.37 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6327, 6330, 136, 160, 12, 133} \[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {2 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {n-1}{2},\frac {n+1}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{(1-n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (n^2+6 n+15\right ) x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{(1-n) (n+1) (n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (-n^3-2 n^2+7 n+18\right ) x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac {(n+6) x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{(n+1) (n+3) \left (c-a^2 c x^2\right )^{5/2}} \]
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Rule 12
Rule 133
Rule 136
Rule 160
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} \, 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 \frac {\left (1-\frac {x}{a}\right )^{-\frac {5}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {5}{2}+\frac {n}{2}}}{x} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {\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 {\left (a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \frac {\left (-\frac {3+n}{a}-\frac {3 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{-\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {5}{2}+\frac {n}{2}}}{x} \, dx,x,\frac {1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {\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 {(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {5}{2}+\frac {n}{2}} \left (\frac {(1+n) (3+n)}{a^2}+\frac {2 (6+n) x}{a^3}\right )}{x} \, dx,x,\frac {1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {\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 {(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {5}{2}+\frac {n}{2}} \left (\frac {(1-n) (1+n) (3+n)}{a^3}-\frac {\left (15+6 n+n^2\right ) x}{a^4}\right )}{x} \, dx,x,\frac {1}{x}\right )}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {\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 {(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (18+7 n-2 n^2-n^3\right ) \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 {\left (a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \frac {\left (9-10 n^2+n^4\right ) \left (1-\frac {x}{a}\right )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}}}{a^4 x} \, dx,x,\frac {1}{x}\right )}{(1-n) (3-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {\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 {(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (18+7 n-2 n^2-n^3\right ) \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 {\left (\left (9-10 n^2+n^4\right ) \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}}}{x} \, dx,x,\frac {1}{x}\right )}{(1-n) (3-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {\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 {(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (18+7 n-2 n^2-n^3\right ) \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 {2 \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 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{(1-n) \left (c-a^2 c x^2\right )^{5/2}} \\ \end{align*}
Time = 2.45 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.43 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\left (-1+a^2 x^2\right ) \left (\frac {8 e^{n \coth ^{-1}(a x)} (n-a x)}{-1+n^2}+\frac {e^{n \coth ^{-1}(a x)} \left (26 n-2 n^3-27 a x+3 a n^2 x+2 n \left (-1+n^2\right ) \cosh \left (2 \coth ^{-1}(a x)\right )-3 a \left (-1+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (3 \coth ^{-1}(a x)\right )\right )}{9-10 n^2+n^4}-\frac {8 a e^{(1+n) \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )}{1+n}\right )}{4 a^5 c \left (c-a^2 c x^2\right )^{3/2}} \]
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\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x^{4}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{4} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^4\,{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]
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