Optimal. Leaf size=83 \[ \frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+1) \sqrt{1-a^2 x^2}}-\frac{a x^{m+2} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.177692, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6153, 6150, 43} \[ \frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+1) \sqrt{1-a^2 x^2}}-\frac{a x^{m+2} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{-\tanh ^{-1}(a x)} x^m \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int x^m (1-a x) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (x^m-a x^{1+m}\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{x^{1+m} \sqrt{c-a^2 c x^2}}{(1+m) \sqrt{1-a^2 x^2}}-\frac{a x^{2+m} \sqrt{c-a^2 c x^2}}{(2+m) \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0447579, size = 50, normalized size = 0.6 \[ \frac{x^{m+1} \sqrt{c-a^2 c x^2} \left (\frac{1}{m+1}-\frac{a x}{m+2}\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 68, normalized size = 0.8 \begin{align*}{\frac{{x}^{1+m} \left ( amx+ax-m-2 \right ) }{ \left ( 2+m \right ) \left ( 1+m \right ) \left ( ax-1 \right ) \left ( ax+1 \right ) }\sqrt{-{a}^{2}c{x}^{2}+c}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04055, size = 85, normalized size = 1.02 \begin{align*} -\frac{{\left (a \sqrt{c}{\left (m + 1\right )} x^{2} - \sqrt{c}{\left (m + 2\right )} x\right )}{\left (a x + 1\right )}{\left (a x - 1\right )} x^{m}}{{\left (m^{2} + 3 \, m + 2\right )} a^{2} x^{2} - m^{2} - 3 \, m - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.65412, size = 166, normalized size = 2. \begin{align*} \frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left ({\left (a m + a\right )} x^{2} -{\left (m + 2\right )} x\right )} x^{m}}{{\left (a^{2} m^{2} + 3 \, a^{2} m + 2 \, a^{2}\right )} x^{2} - m^{2} - 3 \, m - 2} \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{- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt{- c \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} \sqrt{-a^{2} x^{2} + 1} x^{m}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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