Optimal. Leaf size=71 \[ \frac{a x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-a^2 x^2}}+\frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.127907, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {6160, 6140} \[ \frac{a x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-a^2 x^2}}+\frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6140
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int e^{\tanh ^{-1}(a x)} \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int (1+a x) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^2}{\sqrt{1-a^2 x^2}}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}} x^3}{2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0198959, size = 41, normalized size = 0.58 \[ \frac{x \left (\frac{a x^2}{2}+x\right ) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 42, normalized size = 0.6 \begin{align*}{\frac{{x}^{2} \left ( ax+2 \right ) }{2}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.13777, size = 24, normalized size = 0.34 \begin{align*} -\frac{1}{2} i \, \sqrt{c} x^{2} - \frac{i \, \sqrt{c} x}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87118, size = 117, normalized size = 1.65 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x^{3} + 2 \, x^{2}\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\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{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )} \left (a x + 1\right )}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\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 \frac{{\left (a x + 1\right )} \sqrt{c - \frac{c}{a^{2} x^{2}}} x}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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