Optimal. Leaf size=172 \[ -\frac{c (2 m+3) \sqrt{1-a^2 x^2} x^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{(m+1) (m+2) \sqrt{c-a^2 c x^2}}+\frac{2 a c \sqrt{1-a^2 x^2} x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{(m+2) \sqrt{c-a^2 c x^2}}+\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{m+2} \]
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Rubi [A] time = 0.377894, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6167, 6152, 1809, 808, 365, 364} \[ -\frac{c (2 m+3) \sqrt{1-a^2 x^2} x^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{(m+1) (m+2) \sqrt{c-a^2 c x^2}}+\frac{2 a c \sqrt{1-a^2 x^2} x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{(m+2) \sqrt{c-a^2 c x^2}}+\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{m+2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6152
Rule 1809
Rule 808
Rule 365
Rule 364
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx\\ &=-\left (c \int \frac{x^m (1-a x)^2}{\sqrt{c-a^2 c x^2}} \, dx\right )\\ &=\frac{x^{1+m} \sqrt{c-a^2 c x^2}}{2+m}+\frac{\int \frac{x^m \left (-a^2 c (3+2 m)+2 a^3 c (2+m) x\right )}{\sqrt{c-a^2 c x^2}} \, dx}{a^2 (2+m)}\\ &=\frac{x^{1+m} \sqrt{c-a^2 c x^2}}{2+m}+(2 a c) \int \frac{x^{1+m}}{\sqrt{c-a^2 c x^2}} \, dx-\frac{(c (3+2 m)) \int \frac{x^m}{\sqrt{c-a^2 c x^2}} \, dx}{2+m}\\ &=\frac{x^{1+m} \sqrt{c-a^2 c x^2}}{2+m}+\frac{\left (2 a c \sqrt{1-a^2 x^2}\right ) \int \frac{x^{1+m}}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-a^2 c x^2}}-\frac{\left (c (3+2 m) \sqrt{1-a^2 x^2}\right ) \int \frac{x^m}{\sqrt{1-a^2 x^2}} \, dx}{(2+m) \sqrt{c-a^2 c x^2}}\\ &=\frac{x^{1+m} \sqrt{c-a^2 c x^2}}{2+m}-\frac{c (3+2 m) x^{1+m} \sqrt{1-a^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{(1+m) (2+m) \sqrt{c-a^2 c x^2}}+\frac{2 a c x^{2+m} \sqrt{1-a^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{(2+m) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 0.148672, size = 110, normalized size = 0.64 \[ \frac{x^{m+1} \left (\frac{\sqrt{c-a^2 c x^2} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{\sqrt{1-a^2 x^2}}-\frac{2 \sqrt{c-a c x} F_1\left (m+1;\frac{1}{2},-\frac{1}{2};m+2;-a x,a x\right )}{\sqrt{1-a x}}\right )}{m+1} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.477, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m} \left ( ax-1 \right ) }{ax+1}\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 \frac{\sqrt{-a^{2} c x^{2} + c}{\left (a x - 1\right )} x^{m}}{a x + 1}\,{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^{2} c x^{2} + c}{\left (a x - 1\right )} x^{m}}{a x + 1}, 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 \frac{x^{m} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{a x + 1}\, 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 \frac{\sqrt{-a^{2} c x^{2} + c}{\left (a x - 1\right )} x^{m}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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