Optimal. Leaf size=137 \[ \frac{4 x^m \sqrt{c-a^2 c x^2} \text{Hypergeometric2F1}(1,m+1,m+2,-a x)}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x^m \sqrt{c-a^2 c x^2}}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.243495, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {6192, 6193, 88, 64} \[ \frac{4 x^m \sqrt{c-a^2 c x^2} \, _2F_1(1,m+1;m+2;-a x)}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x^m \sqrt{c-a^2 c x^2}}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 88
Rule 64
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} x^{1+m} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \frac{x^m (-1+a x)^2}{1+a x} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (-3 x^m+a x^{1+m}+\frac{4 x^m}{1+a x}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=-\frac{3 x^m \sqrt{c-a^2 c x^2}}{a (1+m) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{1+m} \sqrt{c-a^2 c x^2}}{(2+m) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\left (4 \sqrt{c-a^2 c x^2}\right ) \int \frac{x^m}{1+a x} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=-\frac{3 x^m \sqrt{c-a^2 c x^2}}{a (1+m) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{1+m} \sqrt{c-a^2 c x^2}}{(2+m) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 x^m \sqrt{c-a^2 c x^2} \, _2F_1(1,1+m;2+m;-a x)}{a (1+m) \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.051405, size = 75, normalized size = 0.55 \[ \frac{x^m \sqrt{c-a^2 c x^2} (4 (m+2) \text{Hypergeometric2F1}(1,m+1,m+2,-a x)+m (a x-3)+a x-6)}{a (m+1) (m+2) \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.357, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{-{a}^{2}c{x}^{2}+c} \left ({\frac{ax-1}{ax+1}} \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 \sqrt{-a^{2} c x^{2} + c} x^{m} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{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^{2} c x^{2} + c}{\left (a x - 1\right )} x^{m} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1}, 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 \sqrt{-a^{2} c x^{2} + c} x^{m} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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