Optimal. Leaf size=116 \[ \frac{32 \left (c-a^2 c x^2\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{5-n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-5}{2}} \text{Hypergeometric2F1}\left (5,\frac{5-n}{2},\frac{7-n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a^4 (5-n) x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.213657, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6192, 6195, 131} \[ \frac{32 \left (c-a^2 c x^2\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{5-n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-5}{2}} \, _2F_1\left (5,\frac{5-n}{2};\frac{7-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a^4 (5-n) x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6195
Rule 131
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac{\left (c-a^2 c x^2\right )^{3/2} \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3 \, dx}{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{\frac{3}{2}+\frac{n}{2}}}{x^5} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{32 \left (1-\frac{1}{a x}\right )^{\frac{5-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-5+n)} \left (c-a^2 c x^2\right )^{3/2} \, _2F_1\left (5,\frac{5-n}{2};\frac{7-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a^4 (5-n) \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}\\ \end{align*}
Mathematica [B] time = 2.22317, size = 280, normalized size = 2.41 \[ \frac{c^2 \left (96 a^3 c x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (2 (n-1) e^{(n+1) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )+a x \sqrt{1-\frac{1}{a^2 x^2}} (a x+n) e^{n \coth ^{-1}(a x)}\right )-c \left (a^2 x^2-1\right ) \left (16 a \left (n^3-n^2+3 n-3\right ) x \sqrt{1-\frac{1}{a^2 x^2}} e^{(n+1) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )+2 \left (a^2 x^2-1\right )^2 e^{n \coth ^{-1}(a x)} \left (a \left (n^2+3\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )-a \left (n^2-21\right ) x+2 n \left (\left (n^2+3\right ) \cosh \left (2 \coth ^{-1}(a x)\right )-n^2+1\right )\right )\right )\right )}{192 a \left (c-a^2 c x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.306, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{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{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{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 (-{\left (a^{2} c x^{2} - c\right )} \sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}, 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{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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