Optimal. Leaf size=105 \[ -\frac{c 2^{\frac{n+5}{2}} \sqrt{c-a^2 c x^2} (1-a x)^{\frac{5-n}{2}} \text{Hypergeometric2F1}\left (\frac{1}{2} (-n-3),\frac{5-n}{2},\frac{7-n}{2},\frac{1}{2} (1-a x)\right )}{a (5-n) \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.112971, antiderivative size = 105, 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 = {6143, 6140, 69} \[ -\frac{c 2^{\frac{n+5}{2}} \sqrt{c-a^2 c x^2} (1-a x)^{\frac{5-n}{2}} \, _2F_1\left (\frac{1}{2} (-n-3),\frac{5-n}{2};\frac{7-n}{2};\frac{1}{2} (1-a x)\right )}{a (5-n) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 69
Rubi steps
\begin{align*} \int e^{n \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int e^{n \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int (1-a x)^{\frac{3}{2}-\frac{n}{2}} (1+a x)^{\frac{3}{2}+\frac{n}{2}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{2^{\frac{5+n}{2}} c (1-a x)^{\frac{5-n}{2}} \sqrt{c-a^2 c x^2} \, _2F_1\left (\frac{1}{2} (-3-n),\frac{5-n}{2};\frac{7-n}{2};\frac{1}{2} (1-a x)\right )}{a (5-n) \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0741913, size = 102, normalized size = 0.97 \[ \frac{c 2^{\frac{n+5}{2}} \sqrt{c-a^2 c x^2} (1-a x)^{\frac{5}{2}-\frac{n}{2}} \text{Hypergeometric2F1}\left (-\frac{n}{2}-\frac{3}{2},\frac{5}{2}-\frac{n}{2},\frac{7}{2}-\frac{n}{2},\frac{1}{2}-\frac{a x}{2}\right )}{a (n-5) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.21, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \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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