Optimal. Leaf size=74 \[ \frac{x^2 \sqrt{c-a^2 c x^2}}{2 \sqrt{1-a^2 x^2}}-\frac{a x^3 \sqrt{c-a^2 c x^2}}{3 \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.130159, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6153, 6150, 43} \[ \frac{x^2 \sqrt{c-a^2 c x^2}}{2 \sqrt{1-a^2 x^2}}-\frac{a x^3 \sqrt{c-a^2 c x^2}}{3 \sqrt{1-a^2 x^2}} \]
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 \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{-\tanh ^{-1}(a x)} x \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int x (1-a x) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (x-a x^2\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{x^2 \sqrt{c-a^2 c x^2}}{2 \sqrt{1-a^2 x^2}}-\frac{a x^3 \sqrt{c-a^2 c x^2}}{3 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0218962, size = 42, normalized size = 0.57 \[ -\frac{x^2 (2 a x-3) \sqrt{c-a^2 c x^2}}{6 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 51, normalized size = 0.7 \begin{align*}{\frac{{x}^{2} \left ( 2\,ax-3 \right ) }{ \left ( 6\,ax-6 \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 = 1.00856, size = 55, normalized size = 0.74 \begin{align*} -\frac{{\left (2 \, a \sqrt{c} x^{3} - 3 \, \sqrt{c} x^{2}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )}}{6 \,{\left (a^{2} x^{2} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39566, size = 104, normalized size = 1.41 \begin{align*} \frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x^{3} - 3 \, x^{2}\right )}}{6 \,{\left (a^{2} x^{2} - 1\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 \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}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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