Optimal. Leaf size=244 \[ -\frac{2^{\frac{n-1}{2}} \left (n^2+3\right ) \sqrt{c-a^2 c x^2} (1-a x)^{\frac{3-n}{2}} \text{Hypergeometric2F1}\left (\frac{1}{2} (-n-1),\frac{3-n}{2},\frac{5-n}{2},\frac{1}{2} (1-a x)\right )}{3 a^3 (3-n) \sqrt{1-a^2 x^2}}-\frac{n \sqrt{c-a^2 c x^2} (a x+1)^{\frac{n+3}{2}} (1-a x)^{\frac{3-n}{2}}}{12 a^3 \sqrt{1-a^2 x^2}}-\frac{x \sqrt{c-a^2 c x^2} (a x+1)^{\frac{n+3}{2}} (1-a x)^{\frac{3-n}{2}}}{4 a^2 \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.312724, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6153, 6150, 90, 80, 69} \[ -\frac{2^{\frac{n-1}{2}} \left (n^2+3\right ) \sqrt{c-a^2 c x^2} (1-a x)^{\frac{3-n}{2}} \, _2F_1\left (\frac{1}{2} (-n-1),\frac{3-n}{2};\frac{5-n}{2};\frac{1}{2} (1-a x)\right )}{3 a^3 (3-n) \sqrt{1-a^2 x^2}}-\frac{n \sqrt{c-a^2 c x^2} (a x+1)^{\frac{n+3}{2}} (1-a x)^{\frac{3-n}{2}}}{12 a^3 \sqrt{1-a^2 x^2}}-\frac{x \sqrt{c-a^2 c x^2} (a x+1)^{\frac{n+3}{2}} (1-a x)^{\frac{3-n}{2}}}{4 a^2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 90
Rule 80
Rule 69
Rubi steps
\begin{align*} \int e^{n \tanh ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{n \tanh ^{-1}(a x)} x^2 \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int x^2 (1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2}+\frac{n}{2}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{x (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{3+n}{2}} \sqrt{c-a^2 c x^2}}{4 a^2 \sqrt{1-a^2 x^2}}-\frac{\sqrt{c-a^2 c x^2} \int (1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2}+\frac{n}{2}} (-1-a n x) \, dx}{4 a^2 \sqrt{1-a^2 x^2}}\\ &=-\frac{n (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{3+n}{2}} \sqrt{c-a^2 c x^2}}{12 a^3 \sqrt{1-a^2 x^2}}-\frac{x (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{3+n}{2}} \sqrt{c-a^2 c x^2}}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (\left (3+n^2\right ) \sqrt{c-a^2 c x^2}\right ) \int (1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2}+\frac{n}{2}} \, dx}{12 a^2 \sqrt{1-a^2 x^2}}\\ &=-\frac{n (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{3+n}{2}} \sqrt{c-a^2 c x^2}}{12 a^3 \sqrt{1-a^2 x^2}}-\frac{x (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{3+n}{2}} \sqrt{c-a^2 c x^2}}{4 a^2 \sqrt{1-a^2 x^2}}-\frac{2^{\frac{1}{2} (-1+n)} \left (3+n^2\right ) (1-a x)^{\frac{3-n}{2}} \sqrt{c-a^2 c x^2} \, _2F_1\left (\frac{1}{2} (-1-n),\frac{3-n}{2};\frac{5-n}{2};\frac{1}{2} (1-a x)\right )}{3 a^3 (3-n) \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.191262, size = 135, normalized size = 0.55 \[ \frac{\sqrt{c-a^2 c x^2} (1-a x)^{\frac{3}{2}-\frac{n}{2}} \left (2^{\frac{n+3}{2}} \left (n^2+3\right ) \text{Hypergeometric2F1}\left (\frac{1}{2} (-n-1),\frac{3-n}{2},\frac{5-n}{2},\frac{1}{2} (1-a x)\right )-(n-3) (a x+1)^{\frac{n+3}{2}} (3 a x+n)\right )}{12 a^3 (n-3) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.207, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{x}^{2}\sqrt{-{a}^{2}c{x}^{2}+c}\, 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 \sqrt{-a^{2} c x^{2} + c} x^{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 (\sqrt{-a^{2} c x^{2} + c} x^{2} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname{atanh}{\left (a x \right )}}\, dx \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 \sqrt{-a^{2} c x^{2} + c} x^{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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