Optimal. Leaf size=245 \[ -\frac{3 a^2 x^3 \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n+3}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{\frac{4-n}{2}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+3}{2},\frac{3}{2},\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{2 \left (n^2+8 n+15\right ) (c-a c x)^{7/2}}+\frac{3 a^2 x^3 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{\frac{4-n}{2}}}{2 \left (n^2+8 n+15\right ) (c-a c x)^{7/2}}-\frac{a x^2 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{\frac{2-n}{2}}}{(n+5) (c-a c x)^{7/2}} \]
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Rubi [A] time = 0.270551, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6176, 6181, 94, 132} \[ -\frac{3 a^2 x^3 \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n+3}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{\frac{4-n}{2}} \, _2F_1\left (\frac{1}{2},\frac{n+3}{2};\frac{3}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{2 \left (n^2+8 n+15\right ) (c-a c x)^{7/2}}+\frac{3 a^2 x^3 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{\frac{4-n}{2}}}{2 \left (n^2+8 n+15\right ) (c-a c x)^{7/2}}-\frac{a x^2 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{\frac{2-n}{2}}}{(n+5) (c-a c x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 132
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{7/2} x^{7/2}} \, dx}{(c-a c x)^{7/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{7/2} \operatorname{Subst}\left (\int x^{3/2} \left (1-\frac{x}{a}\right )^{-\frac{7}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a \left (1-\frac{1}{a x}\right )^{\frac{2-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x^2}{(5+n) (c-a c x)^{7/2}}+\frac{\left (3 a \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \sqrt{x} \left (1-\frac{x}{a}\right )^{-\frac{5}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )}{2 (5+n) \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a \left (1-\frac{1}{a x}\right )^{\frac{2-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x^2}{(5+n) (c-a c x)^{7/2}}+\frac{3 a^2 \left (1-\frac{1}{a x}\right )^{\frac{4-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x^3}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}}-\frac{\left (3 a^2 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{\sqrt{x}} \, dx,x,\frac{1}{x}\right )}{4 (3+n) (5+n) \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a \left (1-\frac{1}{a x}\right )^{\frac{2-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x^2}{(5+n) (c-a c x)^{7/2}}+\frac{3 a^2 \left (1-\frac{1}{a x}\right )^{\frac{4-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x^3}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}}-\frac{3 a^2 \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{3+n}{2}} \left (1-\frac{1}{a x}\right )^{\frac{4-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x^3 \, _2F_1\left (\frac{1}{2},\frac{3+n}{2};\frac{3}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.157539, size = 138, normalized size = 0.56 \[ \frac{\left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} \left (3 (a x-1)^2 \left (\frac{a x-1}{a x+1}\right )^{\frac{n+1}{2}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+3}{2},\frac{3}{2},\frac{2}{a x+1}\right )+(a x+1) (-3 a x+2 n+9)\right )}{2 a c^3 (n+3) (n+5) (a x-1)^2 \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.339, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -acx+c \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a c x + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{4} x^{4} - 4 \, a^{3} c^{4} x^{3} + 6 \, a^{2} c^{4} x^{2} - 4 \, a c^{4} x + c^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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