Optimal. Leaf size=80 \[ \frac{x^{m+1} \sqrt{c-\frac{c}{a^2 x^2}}}{m \sqrt{1-a^2 x^2}}+\frac{a x^{m+2} \sqrt{c-\frac{c}{a^2 x^2}}}{(m+1) \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.220095, antiderivative size = 80, 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 = {6160, 6150, 43} \[ \frac{x^{m+1} \sqrt{c-\frac{c}{a^2 x^2}}}{m \sqrt{1-a^2 x^2}}+\frac{a x^{m+2} \sqrt{c-\frac{c}{a^2 x^2}}}{(m+1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^m \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int e^{\tanh ^{-1}(a x)} x^{-1+m} \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 x^{-1+m} (1+a x) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \left (x^{-1+m}+a x^m\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^{1+m}}{m \sqrt{1-a^2 x^2}}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}} x^{2+m}}{(1+m) \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0311057, size = 51, normalized size = 0.64 \[ \frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (\frac{a x^{m+1}}{m+1}+\frac{x^m}{m}\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 53, normalized size = 0.7 \begin{align*}{\frac{{x}^{1+m} \left ( axm+m+1 \right ) }{ \left ( 1+m \right ) m}\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 = 1.16039, size = 41, normalized size = 0.51 \begin{align*} \frac{\sqrt{c} x x^{m}}{i \, m + i} - \frac{i \, \sqrt{c} x^{m}}{a m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87706, size = 153, normalized size = 1.91 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a m x^{2} +{\left (m + 1\right )} x\right )} x^{m} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{{\left (a^{2} m^{2} + a^{2} m\right )} x^{2} - m^{2} - m} \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 (-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^{m}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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